$2x'' = y' +y$
$y'' = -x'+x+3t^2$
I've always learned how to transform this to a system of first-order ODEs (downwards) but never up to higher differentials. How can I solve this, and how does "either x or y" effect what the solution may be?
[Math] Transform the system into a single fourth-order ODE in either x or y
ordinary differential equations
Best Answer
It's just about algebraic manipulations:
You need to isolate one of $x,y$ from the first equation and substitute it into the second. For example, $y'=2x''-y$, so $y''=2x'''-y'=2x'''-2x''+y$.
Substituting into the second: $$2x'''-2x''+y=y'' = -x'+x+3t^2 \hspace{10pt}\Rightarrow \hspace{10pt} y=-2x'''+2x''-x'+x+3t^2$$ Using the first equation once again, we have: $$\begin{align*}-2x''''+2x'''-x''+x'+6t &= y'=2x''-(-2x'''+2x''-x'+x+3t^2) \\ &= 2x'''+x'-x-3t^2\hspace{10pt}\Rightarrow \hspace{10pt}\end{align*}\\ 2x^{(4)}+x''-x=3t^2+6t$$