# Inverse problem of Eigenvalues in DSolve

For my graduation exam I must prepare system of equations to satisfy some specific conditions. I have solutions, output 2, but I need equations eq11 and eq22. So here is an example.

eq11 = (aa1) (X1^\[Prime]\[Prime])[x] + (aa2) X1[x] + (aa3) X2[x];
eq22 = (bb1) (X2^\[Prime]\[Prime])[x] + (bb2) X2[x] + (bb3) X1[x];

ddm = StateSpaceModel[{eq11 == S1*g1[x],
eq22 == S*g2[x]}, {{X1'[x], 0}, {X1[x], 0}, {X2'[x], 0}, {X2[x],
0}}, {{g1[x], 0}, {g2[x], 0}}, {X1[x], X2[x]}, x]

Map[Composition[Eigenvalues, First, Normal], {ddm}] //
Transpose // MatrixForm

output 1 is

$$\left( \begin{array}{cccc|cc} 0 & -\frac{\text{aa2}}{\text{aa1}} & 0 & -\frac{\text{aa3}}{\text{aa1}} & \frac{\text{S1}}{\text{aa1}} & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -\frac{\text{bb3}}{\text{bb1}} & 0 & -\frac{\text{bb2}}{\text{bb1}} & 0 & \frac{S}{\text{bb1}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{array} \right)_{}^{\mathcal{S}}$$

output 2 $$\left( \begin{array}{c} -\frac{\sqrt{-\frac{\text{aa2}}{\text{aa1}}-\frac{\text{bb2}}{\text{bb1}}-\frac{\sqrt{\text{aa2}^2 \text{bb1}^2-2 \text{aa1} \text{aa2} \text{bb1} \text{bb2}+\text{aa1}^2 \text{bb2}^2+4 \text{aa1} \text{aa3} \text{bb1} \text{bb3}}}{\text{aa1} \text{bb1}}}}{\sqrt{2}} \\ \frac{\sqrt{-\frac{\text{aa2}}{\text{aa1}}-\frac{\text{bb2}}{\text{bb1}}-\frac{\sqrt{\text{aa2}^2 \text{bb1}^2-2 \text{aa1} \text{aa2} \text{bb1} \text{bb2}+\text{aa1}^2 \text{bb2}^2+4 \text{aa1} \text{aa3} \text{bb1} \text{bb3}}}{\text{aa1} \text{bb1}}}}{\sqrt{2}} \\ -\sqrt{-\frac{\text{aa2}}{2 \text{aa1}}-\frac{\text{bb2}}{2 \text{bb1}}+\frac{\sqrt{(\text{aa2} \text{bb1}+\text{aa1} \text{bb2})^2-4 \text{aa1} \text{bb1} (\text{aa2} \text{bb2}-\text{aa3} \text{bb3})}}{2 \text{aa1} \text{bb1}}} \\ \sqrt{-\frac{\text{aa2}}{2 \text{aa1}}-\frac{\text{bb2}}{2 \text{bb1}}+\frac{\sqrt{(\text{aa2} \text{bb1}+\text{aa1} \text{bb2})^2-4 \text{aa1} \text{bb1} (\text{aa2} \text{bb2}-\text{aa3} \text{bb3})}}{2 \text{aa1} \text{bb1}}} \end{array} \right)$$

Is it possible to do inverse in $Mathematica$. We have output 2 and I need started equations, because I need to change something in solutions, to put different?

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