N3m - Programming Euler’s method. Implement Euler’s method using technology.


Example 1

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 1)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= -2 \, t y + 4 \, x y - 1\hspace{2em}x( -2 )= -2\]

\[y'= -2 \, x^{2} y + t y^{2} + 1\hspace{2em}y( -2 )= -2\]

Answer.

\(x(-1.4)\approx -1.357\) and \(y(-1.4)\approx -0.1167\).


Example 2

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 2)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -4 \, t^{2} x - t^{2} y - 3\hspace{2em}x( -1 )= 1\]

\[y'= t^{2} y^{2} + t x\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.40)\approx -0.6627\) and \(y(-0.40)\approx -1.231\).


Example 3

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 3)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -3 \, t^{2} x^{2} - 3 \, t^{2} y - 1\hspace{2em}x( 2 )= 0\]

\[y'= 4 \, x^{2} y^{2} + 2 \, t^{2} y - 2\hspace{2em}y( 2 )= -2\]

Answer.

\(x(2.3)\approx 1.283\) and \(y(2.3)\approx -1.727\).


Example 4

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 4)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y^{2} - 3 \, t x + 1\hspace{2em}x( 0 )= 2\]

\[y'= -t^{2} x - 2 \, t y - 1\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.60)\approx 1.105\) and \(y(0.60)\approx -0.5699\).


Example 5

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 5)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= t x - 3 \, x y\hspace{2em}x( 0 )= 2\]

\[y'= t^{2} x - 4 \, t^{2} y + 1\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.60)\approx 0.2463\) and \(y(0.60)\approx 1.277\).


Example 6

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 6)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 3 \, t y^{2} - 3 \, x y + 3\hspace{2em}x( 1 )= 1\]

\[y'= t^{2} x^{2} - 4 \, t^{2} y^{2} + 2\hspace{2em}y( 1 )= -2\]

Answer.

\(x(1.3)\approx 8.490\) and \(y(1.3)\approx 4.582\).


Example 7

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 7)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t^{2} x^{2} + 3 \, t y\hspace{2em}x( 0 )= -1\]

\[y'= -4 \, t^{2} x + 2 \, t y^{2} - 3\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.30)\approx -1.154\) and \(y(0.30)\approx -1.667\).


Example 8

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 8)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 4 \, t^{2} x + 4 \, x y - 1\hspace{2em}x( 1 )= -2\]

\[y'= 2 \, t^{2} x + t y - 3\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.3)\approx -1.827\) and \(y(1.3)\approx -3.460\).


Example 9

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 9)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -3 \, t^{2} x^{2} - x^{2} y^{2} - 1\hspace{2em}x( 0 )= 1\]

\[y'= 4 \, x^{2} y + 3 \, t y + 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.30)\approx 0.1268\) and \(y(0.30)\approx 3.912\).


Example 10

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 10)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y - 2 \, t y + 3\hspace{2em}x( 0 )= -2\]

\[y'= 3 \, t^{2} x + 4 \, t^{2} y + 1\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.60)\approx -1.751\) and \(y(0.60)\approx 0.3600\).


Example 11

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 11)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= t x^{2} + 3 \, t y^{2} - 3\hspace{2em}x( 2 )= -1\]

\[y'= -t^{2} y^{2} - 4 \, x y^{2} - 2\hspace{2em}y( 2 )= 1\]

Answer.

\(x(2.3)\approx -0.7502\) and \(y(2.3)\approx 0.2308\).


Example 12

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 12)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 2 \, t^{2} y^{2} + 2 \, x^{2} y + 2\hspace{2em}x( 1 )= -2\]

\[y'= -4 \, x^{2} y^{2} + t x^{2} - 2\hspace{2em}y( 1 )= 1\]

Answer.

\(x(1.3)\approx -0.6836\) and \(y(1.3)\approx 0.2074\).


Example 13

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 13)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= t^{2} y - x^{2} y + 3\hspace{2em}x( -2 )= -2\]

\[y'= x^{2} y^{2} + t y^{2}\hspace{2em}y( -2 )= -2\]

Answer.

\(x(-1.7)\approx -1.367\) and \(y(-1.7)\approx -1.263\).


Example 14

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 14)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, x^{2} y - 4 \, t y^{2} - 1\hspace{2em}x( -1 )= 2\]

\[y'= t x + 4 \, t y + 2\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.70)\approx 5.716\) and \(y(-0.70)\approx 0.08035\).


Example 15

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 15)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 3 \, t x^{2} + t y - 1\hspace{2em}x( -1 )= 2\]

\[y'= -2 \, t x^{2} + 4 \, x^{2} y - 3\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.70)\approx 1.110\) and \(y(-0.70)\approx -6.628\).


Example 16

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 16)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= x^{2} y^{2} + t y + 2\hspace{2em}x( -1 )= 1\]

\[y'= 2 \, t^{2} y^{2} + 2 \, x y^{2} + 2\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.70)\approx 1.620\) and \(y(-0.70)\approx 0.7901\).


Example 17

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 17)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, x^{2} y + 2 \, t y^{2} - 2\hspace{2em}x( -1 )= -2\]

\[y'= t^{2} y^{2} + 3 \, x y + 3\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.70)\approx -1.968\) and \(y(-0.70)\approx 0.4260\).


Example 18

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 18)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= -3 \, t^{2} y - 4 \, t x\hspace{2em}x( -2 )= 0\]

\[y'= 2 \, t x^{2} - 4 \, x y - 2\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.7)\approx 1.978\) and \(y(-1.7)\approx -0.8414\).


Example 19

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 19)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -2 \, x^{2} y^{2} + 2 \, t x - 3\hspace{2em}x( 0 )= 2\]

\[y'= 3 \, t^{2} y^{2} - x^{2} y^{2} + 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.30)\approx 0.2759\) and \(y(0.30)\approx -1.106\).


Example 20

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 20)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 3 \, t^{2} y^{2} - 4 \, x^{2} y^{2} - 3\hspace{2em}x( 0 )= 0\]

\[y'= -t^{2} y + x y^{2} + 1\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.30)\approx -1.555\) and \(y(0.30)\approx 1.088\).


Example 21

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 21)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y^{2} - t x + 2\hspace{2em}x( 2 )= 1\]

\[y'= -2 \, t x^{2} - x y - 2\hspace{2em}y( 2 )= -2\]

Answer.

\(x(2.6)\approx 0.2049\) and \(y(2.6)\approx -3.063\).


Example 22

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 22)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -x^{2} y + 4 \, t x + 3\hspace{2em}x( 0 )= 0\]

\[y'= -2 \, t^{2} y^{2} + 3 \, t^{2} x + 2\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.60)\approx 1.590\) and \(y(0.60)\approx 2.007\).


Example 23

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 23)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= -t x^{2} - 4 \, x y + 3\hspace{2em}x( 1 )= 2\]

\[y'= -3 \, t x - 3 \, t y - 3\hspace{2em}y( 1 )= -2\]

Answer.

\(x(1.3)\approx 13.13\) and \(y(1.3)\approx -5.796\).


Example 24

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 24)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 2 \, t^{2} y^{2} - 4 \, x^{2} y^{2} + 2\hspace{2em}x( 1 )= -1\]

\[y'= -3 \, x^{2} y + 2 \, t y - 3\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.3)\approx -0.2153\) and \(y(1.3)\approx -1.094\).


Example 25

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 25)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, x y^{2} + 4 \, t y - 1\hspace{2em}x( -1 )= 2\]

\[y'= 2 \, t x^{2} - 2 \, x y^{2} - 3\hspace{2em}y( -1 )= 2\]

Answer.

\(x(-0.70)\approx 4.729\) and \(y(-0.70)\approx -4.399\).


Example 26

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 26)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

\[x'= 4 \, x^{2} y^{2} + 3 \, t y^{2} + 1\hspace{2em}x( -2 )= -2\]

\[y'= -2 \, t^{2} y^{2} - 4 \, x^{2} y^{2} + 3\hspace{2em}y( -2 )= 2\]

Answer.

\(x(-1.4)\approx -0.8088\) and \(y(-1.4)\approx 0.6344\).


Example 27

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 27)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= x y^{2} - 4 \, t y + 1\hspace{2em}x( 1 )= -2\]

\[y'= -3 \, t^{2} x^{2} - 4 \, x y + 2\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.3)\approx -10.36\) and \(y(1.3)\approx -13.47\).


Example 28

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 28)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x - 4 \, t y^{2}\hspace{2em}x( -2 )= 0\]

\[y'= -4 \, x^{2} y + 2 \, t y - 3\hspace{2em}y( -2 )= 1\]

Answer.

\(x(-1.7)\approx 2.456\) and \(y(-1.7)\approx -0.1348\).


Example 29

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 29)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -4 \, t^{2} y^{2} - x y^{2}\hspace{2em}x( 0 )= 2\]

\[y'= 4 \, t^{2} y^{2} + 2 \, x y - 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.30)\approx 0.04812\) and \(y(0.30)\approx 3.033\).


Example 30

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 30)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= 2 \, t x^{2} + t y - 2\hspace{2em}x( -1 )= 1\]

\[y'= 4 \, t x - 4 \, t y - 1\hspace{2em}y( -1 )= -2\]

Answer.

\(x(-0.70)\approx 1.044\) and \(y(-0.70)\approx -7.591\).


Example 31

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 31)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -4 \, x^{2} y^{2} + t x + 3\hspace{2em}x( -1 )= 0\]

\[y'= 2 \, x^{2} y^{2} - 2 \, t^{2} y - 3\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.70)\approx 0.7309\) and \(y(-0.70)\approx -0.7151\).


Example 32

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 32)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= 3 \, t x^{2} + t^{2} y - 3\hspace{2em}x( 0 )= -1\]

\[y'= -4 \, x^{2} y^{2} + 2 \, t x^{2} + 3\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.60)\approx -1.475\) and \(y(0.60)\approx 0.7611\).


Example 33

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 33)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= 2 \, t y^{2} + 4 \, t x + 1\hspace{2em}x( 0 )= 1\]

\[y'= -4 \, x^{2} y - t y + 1\hspace{2em}y( 0 )= 0\]

Answer.

\(x(0.60)\approx 2.949\) and \(y(0.60)\approx 0.03386\).


Example 34

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 34)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 4 \, t^{2} x - x y + 3\hspace{2em}x( 1 )= 0\]

\[y'= -2 \, t x + 4 \, t y - 3\hspace{2em}y( 1 )= -1\]

Answer.

\(x(1.3)\approx 4.159\) and \(y(1.3)\approx -6.837\).


Example 35

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 35)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= 2 \, t^{2} x^{2} + 3 \, t^{2} y - 1\hspace{2em}x( -1 )= 0\]

\[y'= -4 \, x^{2} y^{2} + 2 \, t y^{2} + 1\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.40)\approx 0.2896\) and \(y(-0.40)\approx 0.8183\).


Example 36

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 36)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

\[x'= -2 \, x^{2} y^{2} - 4 \, t^{2} x\hspace{2em}x( -1 )= 2\]

\[y'= 3 \, x y^{2} + 3 \, t y + 2\hspace{2em}y( -1 )= 0\]

Answer.

\(x(-0.40)\approx 0.4277\) and \(y(-0.40)\approx 1.001\).


Example 37

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 37)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= -t^{2} x^{2} + 4 \, t y^{2} - 1\hspace{2em}x( 2 )= 2\]

\[y'= 4 \, t^{2} x - 3 \, t^{2} y\hspace{2em}y( 2 )= 2\]

Answer.

\(x(2.3)\approx 24.13\) and \(y(2.3)\approx 19.08\).


Example 38

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 38)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -t^{2} x^{2} + 3 \, t^{2} y + 1\hspace{2em}x( 1 )= 0\]

\[y'= x^{2} y + t y^{2} + 3\hspace{2em}y( 1 )= -2\]

Answer.

\(x(1.6)\approx -12.89\) and \(y(1.6)\approx -3.994\).


Example 39

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 39)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -3 \, t y^{2} - 4 \, t x - 2\hspace{2em}x( 0 )= -1\]

\[y'= -2 \, t^{2} y^{2} - 2 \, t^{2} x + 3\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.60)\approx -4.541\) and \(y(0.60)\approx 3.043\).


Example 40

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 40)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

\[x'= -t^{2} y^{2} + 3 \, x^{2} y + 1\hspace{2em}x( -2 )= 0\]

\[y'= 3 \, t^{2} x^{2} + 4 \, x y\hspace{2em}y( -2 )= 0\]

Answer.

\(x(-1.7)\approx 0.3002\) and \(y(-1.7)\approx 0.08715\).


Example 41

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 41)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(1.3)\) and \(y(1.3)\) given the following system of IVPs.

\[x'= 3 \, x^{2} y^{2} - t^{2} x - 1\hspace{2em}x( 1 )= 0\]

\[y'= -4 \, t^{2} x + 3 \, x^{2} y + 1\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.3)\approx -0.2426\) and \(y(1.3)\approx 0.5265\).


Example 42

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 42)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t^{2} y^{2} - 2 \, x y^{2}\hspace{2em}x( 0 )= -2\]

\[y'= -t^{2} y^{2} - 2 \, t^{2} x - 1\hspace{2em}y( 0 )= -1\]

Answer.

\(x(0.30)\approx -0.8581\) and \(y(0.30)\approx -1.293\).


Example 43

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 43)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -t x^{2} + 4 \, x y\hspace{2em}x( 1 )= -1\]

\[y'= -2 \, t^{2} x^{2} + 2 \, x^{2} y^{2} - 1\hspace{2em}y( 1 )= 0\]

Answer.

\(x(1.6)\approx -0.2420\) and \(y(1.6)\approx -1.360\).


Example 44

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 44)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

\[x'= -t^{2} y^{2} - t x - 3\hspace{2em}x( 1 )= -1\]

\[y'= 4 \, t^{2} x - 4 \, t y + 3\hspace{2em}y( 1 )= 1\]

Answer.

\(x(1.6)\approx -2.804\) and \(y(1.6)\approx -2.313\).


Example 45

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 45)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -x^{2} y + t y + 2\hspace{2em}x( -1 )= 2\]

\[y'= x^{2} y^{2} - 3 \, t^{2} y + 2\hspace{2em}y( -1 )= 1\]

Answer.

\(x(-0.70)\approx 1.127\) and \(y(-0.70)\approx 2.264\).


Example 46

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 46)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

\[x'= -2 \, t^{2} y^{2} - 4 \, x^{2} y + 3\hspace{2em}x( -1 )= -1\]

\[y'= 4 \, t^{2} x^{2} + 2 \, t y^{2} + 2\hspace{2em}y( -1 )= -1\]

Answer.

\(x(-0.70)\approx -0.01399\) and \(y(-0.70)\approx -0.3053\).


Example 47

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 47)

Use technology to implement Euler's method with \(h=0.020\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

\[x'= -x^{2} y - t y^{2} - 3\hspace{2em}x( 0 )= 1\]

\[y'= 4 \, x^{2} y^{2} + 2 \, t x - 1\hspace{2em}y( 0 )= -2\]

Answer.

\(x(0.60)\approx -0.7424\) and \(y(0.60)\approx -1.150\).


Example 48

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 48)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(2.3)\) and \(y(2.3)\) given the following system of IVPs.

\[x'= 3 \, x^{2} y + 3 \, t y^{2} + 3\hspace{2em}x( 2 )= -1\]

\[y'= t^{2} x^{2} + t^{2} y^{2}\hspace{2em}y( 2 )= -1\]

Answer.

\(x(2.3)\approx 0.2921\) and \(y(2.3)\approx -0.3078\).


Example 49

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 49)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= 4 \, t x^{2} + 2 \, t^{2} y - 2\hspace{2em}x( 0 )= -1\]

\[y'= -4 \, t^{2} y^{2} + 2 \, x y^{2} - 3\hspace{2em}y( 0 )= 1\]

Answer.

\(x(0.30)\approx -1.318\) and \(y(0.30)\approx -0.08504\).


Example 50

N3m - Programming Euler’s method. Implement Euler’s method using technology. (ver. 50)

Use technology to implement Euler's method with \(h=0.010\) to approximate \(x(0.30)\) and \(y(0.30)\) given the following system of IVPs.

\[x'= -3 \, t^{2} x^{2} - 2 \, x^{2} y^{2}\hspace{2em}x( 0 )= 1\]

\[y'= -2 \, x^{2} y^{2} + t^{2} y + 2\hspace{2em}y( 0 )= 2\]

Answer.

\(x(0.30)\approx 0.3189\) and \(y(0.30)\approx 1.939\).