Existence and uniqueness of $y''(t)=\tan(y(t))+\sqrt[3]{t}$

Question

Given the following ODE: $$y''(t)=\tan(y(t))+\sqrt[3]{t}$$

I need to determine for which initial condition there exists a solution, and if it is unique.

Can I solve the problem using the existence and uniqueness theorem for first order ODE? I haven't leanred about higher order ODE yet.

Answer 1

You can call $\dot{y}(t)=z(t)$ and so $\ddot{y}(t)=\dot z(t)$. Then, consider now that you have a system of first order ODEs, or that your unknown function is two dimensional, by calling it $\vec x(t)=\big(y(t), z(t)\big)$. You can also apply the theorem here, but remember that your initial condition has two values for a given $t_0$: $y(t_0)$ and $z(t_0)=\dot y(t_0)$.

In this case, you would have$$\dot{\vec x}(t)=F\big(t, \vec x(t)\big),$$where $\dot{\vec x}(t)=\big(\dot y(t), \dot z(t)\big)$ and $$F\big(t, \vec x\big)=F\big(t,y,z\big)=\big(z,\tan (y)+\sqrt[3]t\big).$$ Where is it continuous in $t$ (or where not)? Where is it Lipschitz in $\vec x$?

More information in this answer and many others in this website: https://math.stackexchange.com/a/1664979/481187

Answer 2

Yes, you can apply the theory of first order ODEs as soon as you have rewritten your equation. See here How to reduce higher order linear ODE to a system of first order ODE? how to do it.