Question

Required information Solve a system of ODEs using Euler's method. Consider the following pair of ODEs over the interval from t= 0 to 0.4 using a step size of 0.25. The initial conditions are 10) = 2 and Z(O) = 4. dy dt = -2y + 4et dz yz? = 32 dt Use the Euler method and write a program to solve this. You do not need to submit the program. (Round the final answers to four decimal places.) The result obtained by solving the equations are as follows: t 0 y 2.0000 1.8551 % Z 4.0000 0.3 2.0708 0.40 1.8034 1.8719

The step size is actually 0.075 not 0.25. Thanks!

Answer 1

Let $X = \begin{bmatrix} y \\ z \end{bmatrix}$ and $F(t, X) = \begin{bmatrix} -2y + 4 e^{-t}\\ -\frac{yz^2}{3} \end{bmatrix}$ . We have $X(0) = \begin{bmatrix} 2 \\ 4\end{bmatrix}$ . According to Euler method,

Xn+1 = Xn + F(tn, Xn)

$= \begin{bmatrix} y_n \\ z_n \end{bmatrix} + 0.075 \begin{bmatrix} -2 y _n + 4e^{-t_n}\\ - \frac{y_ n z_n ^2 }{3} \end{bmatrix}$ .

Note that since $\frac4 {0.075}$ is not an integer, the value of cannot be calculated properly. We must use a extrapolation.

0.4500| 0.0750 2.0000 3.2000 2.0000 4.0000 0.1500 1.9733 2.6880 0.2250 1.9393 2.3306 0.3000 1.8884 2.0672 0.3750 1.8274 1.8655 1.7594 1.7065

First row T, second row y, third row z. With the extrapolation we get and .