# Replace the second-order ODE as two first-order ODEs and use Runge-Kutta method to solve

Replace the following second-order ODE by a system of two first-order ODEs. After writing the two first-order ODEs with the proper initial conditions, compute the corresponding values by using fourth-order Runge-Kutta method for $t = 10 s$ You can take $\Delta t=1$

$y''=\frac{10000}{100-5t}-9.8$

$y(0)=0$

$y'(0)=0$

My work:

Write two first-order ODEs:

$y_1'= y_2$

$y_2'= \frac{10000}{100-5t_1}-9.8$

I am stuck at solving these equations with Runge-Kutta and not sure if $y_1'$ and $y_2'$ are correct. Can you guide me to the right direction?

• It is correct, except you have a $t_1$ instead of $t$. Also, you need $y_1(0) = y'(0)$, $y_2(0) = y(0)$. – copper.hat Mar 6 '13 at 17:40
• Just consider Runge-Kutta working on the vector $(y_1, y_2)$ instead of a scalar. – vonbrand Mar 6 '13 at 18:31