solve coupled second-order differential equation

Question

How can I solve the following set of coupled 2nd order differential equations? \begin{equation} \ddot{x}_{1}= -a^{2}x_{1}+ b^{2}x_{2} \end{equation} \begin{equation} \ddot{x}_{2}= b^{2}x_{1}-c^{2}x_{2} \end{equation} How can I solve for ${x}_{1}$ and ${x}_{2}$ in this case? I tried to calculate the eigenvectors and eigenvalues of the matrix of coefficients but I am completely confused with the algebra. I am a beginner in differential equations.

Answer 1

Hint.

With the eigenvectors for

$$ A = \left(\begin{array}{cc}-a^2 & b^2\\b^2 & -c^2\end{array}\right) $$

calling

$$ T = \left( \begin{array}{cc} \frac{c^2-a^2-\sqrt{4 b^4+\left(a^2-c^2\right)^2}}{2 b^2} & 1 \\ \frac{c^2-a^2+\sqrt{4 b^4+\left(a^2-c^2\right)^2}}{2 b^2} & 1 \\ \end{array} \right) $$

and making $X = T^{-1}Y$ we have

$$ T^{-1}\ddot Y = A T^{-1}Y $$

and then

$$ \ddot Y = T A T^{-1} Y = \Lambda Y $$

so the new system reads

$$ \ddot y_1 = \lambda_1 y_1\\ \ddot y_2 = \lambda_2 y_2 $$

Here

$$ \lambda_1 = -\frac 12\left(a^2+c^2+\sqrt{(a^2-c^2)^2-4b^4}\right)\\ \lambda_2 = -\frac 12\left(a^2+c^2-\sqrt{(a^2-c^2)^2-4b^4}\right) $$