Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(\frac{29}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{29}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{3}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.91\) meters from its natural position, and the outer mass is located approximately \(-3.23\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -3 \, \cos\left(2 \, t\right) + 6 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.67\) meters from its natural position, and the outer mass is located approximately \(4.22\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(10\) meters outwards from its natural position, while the outer mass is moved \(\frac{75}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 10 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{75}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) + 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(2 \, \sqrt{5} t\right) + 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(5.63\) meters from its natural position, and the outer mass is located approximately \(8.48\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(10\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -10 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{3} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -2 \, \cos\left(2 \, \sqrt{3} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.490\) meters from its natural position, and the outer mass is located approximately \(-6.95\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(14\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 14 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{3} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 2 \, \cos\left(2 \, \sqrt{3} t\right) + 12 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-6.04\) meters from its natural position, and the outer mass is located approximately \(-9.78\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(5\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(0.555\) meters from its natural position, and the outer mass is located approximately \(-3.55\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{17}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{6} t\right) - 5 \, \cos\left(t\right)\]

\[x_2= \frac{3}{2} \, \cos\left(\sqrt{6} t\right) - 10 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-4.27\) meters from its natural position, and the outer mass is located approximately \(-1.41\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{20}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{20}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{21} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.96\) meters from its natural position, and the outer mass is located approximately \(-6.35\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(9\) meters outwards from its natural position, while the outer mass is moved \(8\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 9 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 8 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{6} t\right) + 5 \, \cos\left(t\right)\]

\[x_2= -2 \, \cos\left(\sqrt{6} t\right) + 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-6.99\) meters from its natural position, and the outer mass is located approximately \(-4.67\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{33}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{33}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.540\) meters from its natural position, and the outer mass is located approximately \(-7.77\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(2\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= 2 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{5} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 6 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(6.39\) meters from its natural position, and the outer mass is located approximately \(-1.53\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{34}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{34}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-7.67\) meters from its natural position, and the outer mass is located approximately \(-10.8\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(\frac{15}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{15}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(\sqrt{10} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.334\) meters from its natural position, and the outer mass is located approximately \(-3.83\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(9\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -9 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{10} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{10} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.0322\) meters from its natural position, and the outer mass is located approximately \(8.92\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{25}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{25}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -3 \, \cos\left(2 \, \sqrt{5} t\right) - \frac{16}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.540\) meters from its natural position, and the outer mass is located approximately \(-4.46\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{33}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{33}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.05\) meters from its natural position, and the outer mass is located approximately \(-13.1\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(9\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -9 ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -5 \, \cos\left(2 \, t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(6.78\) meters from its natural position, and the outer mass is located approximately \(-0.841\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(5\) meters outwards from its natural position, while the outer mass is moved \(\frac{13}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 5 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{13}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{4} \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-4.97\) meters from its natural position, and the outer mass is located approximately \(-3.21\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{47}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{47}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{10}{3} \, \cos\left(\sqrt{15} t\right) - \frac{9}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-7.31\) meters from its natural position, and the outer mass is located approximately \(-0.390\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{10}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{10}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{15} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{8}{3} \, \cos\left(\sqrt{15} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.667\) meters from its natural position, and the outer mass is located approximately \(7.47\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{43}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{43}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.35\) meters from its natural position, and the outer mass is located approximately \(6.12\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(4.64\) meters from its natural position, and the outer mass is located approximately \(8.16\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.95\) meters from its natural position, and the outer mass is located approximately \(-5.24\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{31}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{31}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{21} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{5}{6} \, \cos\left(\sqrt{21} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(5.96\) meters from its natural position, and the outer mass is located approximately \(4.14\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(4\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -4 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{17}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) - 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-3.36\) meters from its natural position, and the outer mass is located approximately \(-4.57\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(4\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -4 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) - 3 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) - 6 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(0.827\) meters from its natural position, and the outer mass is located approximately \(-3.38\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(\frac{35}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{35}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) - 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-4.98\) meters from its natural position, and the outer mass is located approximately \(-9.43\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{7}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{7}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(\sqrt{6} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= \frac{5}{2} \, \cos\left(\sqrt{6} t\right) - 6 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.321\) meters from its natural position, and the outer mass is located approximately \(2.96\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(3.16\) meters from its natural position, and the outer mass is located approximately \(-2.66\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(\frac{31}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{31}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(4.75\) meters from its natural position, and the outer mass is located approximately \(11.1\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{47}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{47}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{10}{3} \, \cos\left(\sqrt{15} t\right) - \frac{9}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-7.31\) meters from its natural position, and the outer mass is located approximately \(-0.390\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(6\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 6 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{6} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-3.92\) meters from its natural position, and the outer mass is located approximately \(-2.99\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, t\right) - 5 \, \cos\left(t\right)\]

\[x_2= 3 \, \cos\left(2 \, t\right) - 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(3.70\) meters from its natural position, and the outer mass is located approximately \(6.10\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.29\) meters from its natural position, and the outer mass is located approximately \(3.51\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{19}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= \frac{19}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= \frac{3}{2} \, \cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.71\) meters from its natural position, and the outer mass is located approximately \(3.69\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(0\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 0 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -4 \, \cos\left(2 \, t\right) + 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-1.89\) meters from its natural position, and the outer mass is located approximately \(-2.03\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-5.95\) meters from its natural position, and the outer mass is located approximately \(-2.93\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(4\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 4 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{3} t\right) + 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{3} t\right) + 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.36\) meters from its natural position, and the outer mass is located approximately \(-5.81\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{15}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-2.54\) meters from its natural position, and the outer mass is located approximately \(9.18\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(15\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 15 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{5} t\right) + 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-7.34\) meters from its natural position, and the outer mass is located approximately \(-14.3\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-4.48\) meters from its natural position, and the outer mass is located approximately \(-4.30\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(10\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= 10 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{5} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 6 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(1.56\) meters from its natural position, and the outer mass is located approximately \(-8.48\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{11}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{11}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{6} t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{2} \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(3.43\) meters from its natural position, and the outer mass is located approximately \(3.23\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(5\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.01\) meters from its natural position, and the outer mass is located approximately \(-4.96\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{22}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{22}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{15} t\right) - 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{4}{3} \, \cos\left(\sqrt{15} t\right) - 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.98\) meters from its natural position, and the outer mass is located approximately \(1.94\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{21} t\right) + 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 9 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.160\) meters from its natural position, and the outer mass is located approximately \(-4.33\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(10\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -10 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -6 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-0.702\) meters from its natural position, and the outer mass is located approximately \(9.11\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{15}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.37\) meters from its natural position, and the outer mass is located approximately \(-6.19\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(12\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 12 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) + 5 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) + 10 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-6.87\) meters from its natural position, and the outer mass is located approximately \(-7.98\) meters from its natural position.

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(\frac{45}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{45}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{3} t\right) - 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{2} \, \cos\left(2 \, \sqrt{3} t\right) - 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.61\) meters from its natural position, and the outer mass is located approximately \(-16.7\) meters from its natural position.