Is there any general and systematic way of reducing the higher order linear ODE to a system of first order ODE?
For example, assume we have $a_3x^{(3)}+a_2x^{(2)}+a_1x^{(1)}+a_0x=0$, then how do we convert this into matrix form(a system of first order ODE). And after we solve the system of equation, how to combine them into our final solution $x(t)$?
Thanks for helping me out!
Best Answer
Starting with $$ x^{(n)}=f(t,x,x',x'',\ldots,x^{(n-1)}), $$ you get the system \begin{align*} x_1&=x\\ x_1'&=x'=x_2\\ x_2'&=x''=x_3\\ &\ldots\\ x_{n-1}'&=x^{(n-1)}=x_{n}\\ x_n^{'}&=x^{(n)}=f(t,x_1,x_2,\ldots,x_{n-1}) \end{align*}