# Transform 2nd order ODE system $y_1''=2y_1+2y_2'-3y_2, y_2''=-2y_1'+5y_2'-4y_2$ into 1st order ODE

Solve

$$\begin{cases}y_1''=2y_1+2y_2'-3y_2\\y_2''=-2y_1'+5y_2'-4y_2\end{cases}$$ With the initial conditions $$y_1(0)=5, y_1'(0)=-1, y_2(0)=-1, y_2'(0)=3$$

The hint is to consider an equivalent 1st order ODE, but I don't know how to find it.

So far I have rewritten everything as a matrix. So we would have

$$y'=Ay+By'$$

where $$A =\left [ \begin{matrix} 2 & -3 \\ 0 & -4 \\ \end{matrix} \right ]$$ and $$B=\left[ \begin{matrix} 0 & 2 \\ -2 & 5 \\ \end{matrix} \right ]$$

I also calculated the eigenvectors of A and B but what do I do with them now and do I need them at all? I also saw examples where everything would be put into a 4x4 matrix, but how would that work? How would I transform everything into a 1st order ODE?

Try to rewrite the system as: $$\pmatrix {y_1 \\y_2 \\y_1'\\y_2'}'=A \pmatrix {y_1 \\y_2 \\y_1'\\y_2'}$$ Where $$A$$ is a $$4 \times 4$$ matrix.
• use the system you are given ...and note that $y'_1=y'_1$ and $y'_2=y'_2$ @NightOwl26 May 21, 2021 at 13:19