N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method.


Example 1

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 1)

Use Euler's method with \(h=0.20\) 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.

\(x(-1.8)\approx -0.6000\) and \(y(-1.8)\approx -0.2000\).

\(x(-1.6)\approx -0.8480\) and \(y(-1.6)\approx 0.01440\).

\(x(-1.4)\approx -1.049\) and \(y(-1.4)\approx 0.2102\).


Example 2

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 2)

Use Euler's method with \(h=0.20\) 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.

\(x(-0.80)\approx 0.0000\) and \(y(-0.80)\approx -1.400\).

\(x(-0.60)\approx -0.4208\) and \(y(-0.60)\approx -1.149\).

\(x(-0.40)\approx -0.8169\) and \(y(-0.40)\approx -1.004\).


Example 3

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 3)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.8)\approx -0.2000\) and \(y(-1.8)\approx -0.6000\).

\(x(-1.6)\approx -2.059\) and \(y(-1.6)\approx -0.5088\).

\(x(-1.4)\approx -11.98\) and \(y(-1.4)\approx 1.325\).


Example 4

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 4)

Use Euler's method with \(h=0.20\) 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.

\(x(0.20)\approx 2.200\) and \(y(0.20)\approx -0.2000\).

\(x(0.40)\approx 1.981\) and \(y(0.40)\approx -0.4016\).

\(x(0.60)\approx 1.199\) and \(y(0.60)\approx -0.6007\).


Example 5

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 5)

Use Euler's method with \(h=0.20\) 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.

\(x(0.20)\approx 0.8000\) and \(y(0.20)\approx 1.200\).

\(x(0.40)\approx 0.2560\) and \(y(0.40)\approx 1.368\).

\(x(0.60)\approx 0.06635\) and \(y(0.60)\approx 1.401\).


Example 6

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 6)

Use Euler's method with \(h=0.20\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

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

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

Answer.

\(x(2.2)\approx -15.00\) and \(y(2.2)\approx 1.800\).

\(x(2.4)\approx -33.55\) and \(y(2.4)\approx -2.120\).

\(x(2.6)\approx -69.76\) and \(y(2.6)\approx -69.41\).


Example 7

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 7)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx -1.000\) and \(y(0.10)\approx -1.300\).

\(x(0.20)\approx -1.035\) and \(y(0.20)\approx -1.562\).

\(x(0.30)\approx -1.112\) and \(y(0.30)\approx -1.748\).


Example 8

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 8)

Use Euler's method with \(h=0.10\) 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.

\(x(1.1)\approx -2.900\) and \(y(1.1)\approx -0.7000\).

\(x(1.2)\approx -3.592\) and \(y(1.2)\approx -1.779\).

\(x(1.3)\approx -3.205\) and \(y(1.3)\approx -3.327\).


Example 9

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 9)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx 0.5000\) and \(y(0.10)\approx 3.000\).

\(x(0.20)\approx 0.1742\) and \(y(0.20)\approx 3.590\).

\(x(0.30)\approx 0.03475\) and \(y(0.30)\approx 4.049\).


Example 10

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 10)

Use Euler's method with \(h=0.20\) 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.

\(x(0.20)\approx -1.400\) and \(y(0.20)\approx 0.2000\).

\(x(0.40)\approx -1.130\) and \(y(0.40)\approx 0.3728\).

\(x(0.60)\approx -0.9698\) and \(y(0.60)\approx 0.5121\).


Example 11

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 11)

Use Euler's method with \(h=0.20\) to approximate \(x(-0.40)\) and \(y(-0.40)\) given the following system of IVPs.

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

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

Answer.

\(x(-0.80)\approx 1.000\) and \(y(-0.80)\approx -0.6000\).

\(x(-0.60)\approx 1.298\) and \(y(-0.60)\approx -1.670\).

\(x(-0.40)\approx 2.499\) and \(y(-0.40)\approx -3.756\).


Example 12

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 12)

Use Euler's method with \(h=0.10\) 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.

\(x(1.1)\approx -0.8000\) and \(y(1.1)\approx -0.4000\).

\(x(1.2)\approx -0.6125\) and \(y(1.2)\approx -0.5706\).

\(x(1.3)\approx -0.3615\) and \(y(1.3)\approx -0.7744\).


Example 13

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 13)

Use Euler's method with \(h=0.10\) 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.

\(x(-1.9)\approx -1.700\) and \(y(-1.9)\approx -1.200\).

\(x(-1.8)\approx -1.486\) and \(y(-1.8)\approx -1.057\).

\(x(-1.7)\approx -1.295\) and \(y(-1.7)\approx -1.012\).


Example 14

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 14)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 3.100\) and \(y(-0.90)\approx 0.6000\).

\(x(-0.80)\approx 4.283\) and \(y(-0.80)\approx 0.3050\).

\(x(-0.70)\approx 5.331\) and \(y(-0.70)\approx 0.06478\).


Example 15

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 15)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 0.8000\) and \(y(-0.90)\approx -2.100\).

\(x(-0.80)\approx 0.7162\) and \(y(-0.80)\approx -2.822\).

\(x(-0.70)\approx 0.7189\) and \(y(-0.70)\approx -3.619\).


Example 16

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 16)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 1.200\) and \(y(-0.90)\approx 0.2000\).

\(x(-0.80)\approx 1.388\) and \(y(-0.80)\approx 0.4161\).

\(x(-0.70)\approx 1.588\) and \(y(-0.70)\approx 0.6863\).


Example 17

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 17)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx -2.200\) and \(y(-0.90)\approx 0.3000\).

\(x(-0.80)\approx -2.126\) and \(y(-0.80)\approx 0.4093\).

\(x(-0.70)\approx -1.983\) and \(y(-0.70)\approx 0.4590\).


Example 18

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 18)

Use Euler's method with \(h=0.10\) 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.

\(x(-1.9)\approx 0.0000\) and \(y(-1.9)\approx -0.2000\).

\(x(-1.8)\approx 0.2166\) and \(y(-1.8)\approx -0.4000\).

\(x(-1.7)\approx 0.7614\) and \(y(-1.7)\approx -0.5822\).


Example 19

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 19)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx 0.9000\) and \(y(0.10)\approx -1.300\).

\(x(0.20)\approx 0.3442\) and \(y(0.20)\approx -1.332\).

\(x(0.30)\approx 0.01596\) and \(y(0.30)\approx -1.232\).


Example 20

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 20)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx -0.3000\) and \(y(0.10)\approx 1.100\).

\(x(0.20)\approx -0.6399\) and \(y(0.20)\approx 1.163\).

\(x(0.30)\approx -1.145\) and \(y(0.30)\approx 1.171\).


Example 21

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 21)

Use Euler's method with \(h=0.20\) to approximate \(x(1.6)\) and \(y(1.6)\) given the following system of IVPs.

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

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

Answer.

\(x(1.2)\approx 0.4000\) and \(y(1.2)\approx 1.200\).

\(x(1.4)\approx 0.3616\) and \(y(1.4)\approx 1.583\).

\(x(1.6)\approx 0.3298\) and \(y(1.6)\approx 2.527\).


Example 22

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 22)

Use Euler's method with \(h=0.20\) 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.

\(x(0.20)\approx 0.6000\) and \(y(0.20)\approx 1.400\).

\(x(0.40)\approx 1.195\) and \(y(0.40)\approx 1.783\).

\(x(0.60)\approx 1.668\) and \(y(0.60)\approx 2.094\).


Example 23

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 23)

Use Euler's method with \(h=0.10\) 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.

\(x(1.1)\approx 3.500\) and \(y(1.1)\approx -2.300\).

\(x(1.2)\approx 5.672\) and \(y(1.2)\approx -2.996\).

\(x(1.3)\approx 8.909\) and \(y(1.3)\approx -4.260\).


Example 24

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 24)

Use Euler's method with \(h=0.10\) 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.

\(x(1.1)\approx -0.8000\) and \(y(1.1)\approx -0.3000\).

\(x(1.2)\approx -0.6013\) and \(y(1.2)\approx -0.6084\).

\(x(1.3)\approx -0.3482\) and \(y(1.3)\approx -0.9884\).


Example 25

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 25)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 2.700\) and \(y(-0.90)\approx -0.7000\).

\(x(-0.80)\approx 3.117\) and \(y(-0.80)\approx -2.577\).

\(x(-0.70)\approx 7.980\) and \(y(-0.70)\approx -8.570\).


Example 26

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 26)

Use Euler's method with \(h=0.20\) to approximate \(x(0.60)\) and \(y(0.60)\) given the following system of IVPs.

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

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

Answer.

\(x(0.20)\approx -0.6000\) and \(y(0.20)\approx 1.400\).

\(x(0.40)\approx -1.016\) and \(y(0.40)\approx 1.332\).

\(x(0.60)\approx -1.289\) and \(y(0.60)\approx 0.5917\).


Example 27

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 27)

Use Euler's method with \(h=0.10\) 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.

\(x(1.1)\approx -1.900\) and \(y(1.1)\approx -1.000\).

\(x(1.2)\approx -1.550\) and \(y(1.2)\approx -2.870\).

\(x(1.3)\approx -1.349\) and \(y(1.3)\approx -5.488\).


Example 28

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 28)

Use Euler's method with \(h=0.10\) 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.

\(x(-1.9)\approx 0.8000\) and \(y(-1.9)\approx 0.3000\).

\(x(-1.8)\approx 1.446\) and \(y(-1.8)\approx -0.1908\).

\(x(-1.7)\approx 2.409\) and \(y(-1.7)\approx -0.2625\).


Example 29

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 29)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx 1.200\) and \(y(0.10)\approx 2.600\).

\(x(0.20)\approx 0.3618\) and \(y(0.20)\approx 3.051\).

\(x(0.30)\approx -0.1239\) and \(y(0.30)\approx 3.221\).


Example 30

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 30)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 0.8000\) and \(y(-0.90)\approx -3.300\).

\(x(-0.80)\approx 0.7818\) and \(y(-0.80)\approx -4.876\).

\(x(-0.70)\approx 0.8741\) and \(y(-0.70)\approx -6.786\).


Example 31

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 31)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 0.3000\) and \(y(-0.90)\approx -0.3000\).

\(x(-0.80)\approx 0.5698\) and \(y(-0.80)\approx -0.5498\).

\(x(-0.70)\approx 0.7849\) and \(y(-0.70)\approx -0.7598\).


Example 32

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 32)

Use Euler's method with \(h=0.20\) to approximate \(x(-1.4)\) and \(y(-1.4)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.8)\approx -5.800\) and \(y(-1.8)\approx 0.0000\).

\(x(-1.6)\approx -16.88\) and \(y(-1.6)\approx -0.6000\).

\(x(-1.4)\approx -38.55\) and \(y(-1.4)\approx 3.080\).


Example 33

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 33)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.9)\approx -3.600\) and \(y(-1.9)\approx -1.200\).

\(x(-1.8)\approx -8.597\) and \(y(-1.8)\approx -4.379\).

\(x(-1.7)\approx -48.83\) and \(y(-1.7)\approx -31.82\).


Example 34

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 34)

Use Euler's method with \(h=0.10\) to approximate \(x(-1.7)\) and \(y(-1.7)\) given the following system of IVPs.

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

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

Answer.

\(x(-1.9)\approx 1.500\) and \(y(-1.9)\approx -0.9000\).

\(x(-1.8)\approx 3.097\) and \(y(-1.8)\approx -0.7010\).

\(x(-1.7)\approx 11.25\) and \(y(-1.7)\approx -0.2930\).


Example 35

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 35)

Use Euler's method with \(h=0.20\) 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.

\(x(-0.80)\approx 0.4000\) and \(y(-0.80)\approx 0.8000\).

\(x(-0.60)\approx 0.5482\) and \(y(-0.60)\approx 0.7133\).

\(x(-0.40)\approx 0.5455\) and \(y(-0.40)\approx 0.6689\).


Example 36

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 36)

Use Euler's method with \(h=0.20\) 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.

\(x(-0.80)\approx 0.4000\) and \(y(-0.80)\approx 0.4000\).

\(x(-0.60)\approx 0.1850\) and \(y(-0.60)\approx 0.6464\).

\(x(-0.40)\approx 0.1260\) and \(y(-0.40)\approx 0.8601\).


Example 37

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 37)

Use Euler's method with \(h=0.10\) 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.

\(x(2.1)\approx 3.500\) and \(y(2.1)\approx 2.800\).

\(x(2.2)\approx 4.583\) and \(y(2.2)\approx 5.270\).

\(x(2.3)\approx 18.75\) and \(y(2.3)\approx 6.492\).


Example 38

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 38)

Use Euler's method with \(h=0.20\) 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.

\(x(1.2)\approx -1.000\) and \(y(1.2)\approx -0.6000\).

\(x(1.4)\approx -1.606\) and \(y(1.4)\approx -0.03360\).

\(x(1.6)\approx -2.457\) and \(y(1.6)\approx 0.5494\).


Example 39

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 39)

Use Euler's method with \(h=0.20\) 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.

\(x(0.20)\approx -1.400\) and \(y(0.20)\approx 2.600\).

\(x(0.40)\approx -2.387\) and \(y(0.40)\approx 3.114\).

\(x(0.60)\approx -4.351\) and \(y(0.60)\approx 3.246\).


Example 40

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 40)

Use Euler's method with \(h=0.10\) 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.

\(x(-1.9)\approx 0.1000\) and \(y(-1.9)\approx 0.0000\).

\(x(-1.8)\approx 0.2000\) and \(y(-1.8)\approx 0.01083\).

\(x(-1.7)\approx 0.3001\) and \(y(-1.7)\approx 0.05058\).


Example 41

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 41)

Use Euler's method with \(h=0.10\) to approximate \(x(-0.70)\) and \(y(-0.70)\) given the following system of IVPs.

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

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

Answer.

\(x(-0.90)\approx 2.000\) and \(y(-0.90)\approx -2.800\).

\(x(-0.80)\approx 3.555\) and \(y(-0.80)\approx -5.028\).

\(x(-0.70)\approx 6.460\) and \(y(-0.70)\approx -15.33\).


Example 42

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 42)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx -1.600\) and \(y(0.10)\approx -1.100\).

\(x(0.20)\approx -1.208\) and \(y(0.20)\approx -1.198\).

\(x(0.30)\approx -0.8383\) and \(y(0.30)\approx -1.294\).


Example 43

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 43)

Use Euler's method with \(h=0.20\) to approximate \(x(2.6)\) and \(y(2.6)\) given the following system of IVPs.

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

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

Answer.

\(x(2.2)\approx -1.400\) and \(y(2.2)\approx 1.000\).

\(x(2.4)\approx -1.275\) and \(y(2.4)\approx 2.184\).

\(x(2.6)\approx -0.04759\) and \(y(2.6)\approx 4.846\).


Example 44

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 44)

Use Euler's method with \(h=0.20\) 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.

\(x(1.2)\approx -1.600\) and \(y(1.2)\approx 0.0000\).

\(x(1.4)\approx -1.816\) and \(y(1.4)\approx -1.243\).

\(x(1.6)\approx -2.513\) and \(y(1.6)\approx -2.098\).


Example 45

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 45)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx 1.700\) and \(y(-0.90)\approx 1.300\).

\(x(-0.80)\approx 1.407\) and \(y(-0.80)\approx 1.673\).

\(x(-0.70)\approx 1.142\) and \(y(-0.70)\approx 2.105\).


Example 46

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 46)

Use Euler's method with \(h=0.10\) 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.

\(x(-0.90)\approx -0.5000\) and \(y(-0.90)\approx -0.6000\).

\(x(-0.80)\approx -0.1983\) and \(y(-0.80)\approx -0.3838\).

\(x(-0.70)\approx 0.08886\) and \(y(-0.70)\approx -0.1973\).


Example 47

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 47)

Use Euler's method with \(h=0.20\) 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.

\(x(0.20)\approx 0.8000\) and \(y(0.20)\approx 1.000\).

\(x(0.40)\approx 0.03200\) and \(y(0.40)\approx 1.376\).

\(x(0.60)\approx -0.7197\) and \(y(0.60)\approx 1.183\).


Example 48

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 48)

Use Euler's method with \(h=0.10\) 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.

\(x(2.1)\approx -0.4000\) and \(y(2.1)\approx -0.2000\).

\(x(2.2)\approx -0.08440\) and \(y(2.2)\approx -0.1118\).

\(x(2.3)\approx 0.2236\) and \(y(2.3)\approx -0.1023\).


Example 49

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 49)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx -1.200\) and \(y(0.10)\approx 0.5000\).

\(x(0.20)\approx -1.341\) and \(y(0.20)\approx 0.1390\).

\(x(0.30)\approx -1.396\) and \(y(0.30)\approx -0.1665\).


Example 50

N2 - Euler’s method for IVP systems. Estimate the value of an IVP system solution using Euler’s method. (ver. 50)

Use Euler's method with \(h=0.10\) 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.

\(x(0.10)\approx 0.2000\) and \(y(0.10)\approx 1.400\).

\(x(0.20)\approx 0.1842\) and \(y(0.20)\approx 1.586\).

\(x(0.30)\approx 0.1667\) and \(y(0.30)\approx 1.775\).