# System of ODEs Runge-Kutta method.

I need to solve the following Cauchy problem using Runge Kutta method (do 2 iterations). $$y'(t)=x(t)y(t)+x^2(t)$$ $$x'(t)=y^2(t)$$ $$y(0)=x(0)=1$$ Let $h = 1/2$, then we get $x(1/2) = x(0) + \frac{1}{6}(k_1+2k_2+2k_3+k_4)$
$k_1 = y^2(0) = 1$ $k_2 = y^2(1/4)$ What should I do in this case, when $y^2(1/4)$ is unknown? Find it approximation (for example using Euler method)? Am I missing something?

• You can't apply the RK4 method for a single ODE to system of ODEs. See here for an example. Jun 21, 2017 at 13:16

You have $f(\binom{x}{y})=\binom{y^2}{x^2+xy}$. Give the state vector a different name, $v=\binom{x}{y}$. Then $$k_1=hf(v_0)$$
The vector $k_2$ is computed at the position $v_0+\frac12k_1$, $$k_2=hf(v_0+\tfrac12k_1)=\pmatrix{h(y_0+\tfrac12k_{y1})^2\\h(x_0+\tfrac12k_{x1})^2+h(x_0+\tfrac12k_{x1})(y_0+\tfrac12k_{y1})}$$ etc.
It is useful to have the evaluation of $f$ as a separate function so that one is forced to compute the intermediate positions only once.