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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?

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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.

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  • $\begingroup$ uhhh and how would i do that? $\endgroup$
    – NightOwl26
    May 21, 2021 at 13:12
  • $\begingroup$ use the system you are given ...and note that $y'_1=y'_1$ and $y'_2=y'_2$ @NightOwl26 $\endgroup$ May 21, 2021 at 13:19
  • $\begingroup$ okay so what do i do once i've found the matrix? $\endgroup$
    – NightOwl26
    May 21, 2021 at 13:21
  • $\begingroup$ find the eigenvalues and eigenvectors @NightOwl26 $\endgroup$ May 21, 2021 at 13:23

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