solving third order nonlinear differential equation

Question

Can someone help me in solving : $$ y'''(x)+2\,y(x)y''(x)=0 $$ using Runge-Kutta method? I'm not sure how to linearize it.

Thanks in advance

Answer 1

To apply a RK method, you need not to linearize, but to transform into a first order system. Assign to a state vector $u\in\Bbb R^3$ the intended meaning $u_0=y$, $u_1=y'$, $u_2=y''$, then the ODE function in $u'(t)=F(t,u(t))$ is $$ F(t,u)=\pmatrix{u_1\\u_2\\-2u_0u_2} $$