# How to Solve These Irregular Coupled Second Order Differential Equations

I have a question on how to solve this pair of coupled differential equations. \begin{align*} \omega_1^2 Q_1 &= -\ddot{Q}_1 - \frac{M}{L_1}\ddot{Q}_2 \\ \omega_2^2 Q_2 &= -\ddot{Q}_2 - \frac{M}{L_2}\ddot{Q}_1 \end{align*} for $$Q_1 (t)$$ and $$Q_2 (t)$$. Generally I would solve these types of equations by using a matrix $$M^{-1}K - \omega^2 I = 0$$. However, in this case I don't really see how I can create matrices to solve this problem and I can't find any neat trick to solve these equations. If anyone could help that would be great!

• Maybe you can take the Fourier transformation. It'll give you two linear equations which you can solve for $\tilde{Q}_{1,2}$ and then you can transform back to the time domain. Jul 18, 2020 at 23:15

Let $$l_1=\frac{M}{L_1},l_2=\frac{M}{L_2}$$. \begin{align*} \ \ \ \omega_1^2 Q_1 &= -\ddot{Q}_1 - l_1\ddot{Q}_2 \\ +bc\times(\omega_2^2 Q_2 &= -\ddot{Q}_2 - l_2\ddot{Q}_1)\\ \hline \omega_1^2 Q_1 +bc\omega_2^2 Q_2&=-\left(1+bcl_2\right)\ddot{Q}_1-\left(l_1+bc\right)\ddot{Q}_2\\ \omega_1^2 Q_1 +bc\omega_2^2 Q_2&=-\left(1+bcl_2\right)\ddot{Q}_1-\left(\frac{l_1}c+b\right)c\ddot{Q}_2\\ \star(Q_1 +cQ_2)&=-\star(\ddot{Q}_1+c\ddot{Q}_2)\\ \end{align*} is only possible when \begin{align*} b&=\frac{\omega_1^2}{\omega_2^2}\\ 1+bcl_2=\frac{l_1}c+b\Rightarrow bl_2c^2-(b-1)c-l_1=0\Rightarrow c&=\frac{(b-1)\pm\sqrt{(b-1^2)+4bl_1l_2}}{2bl_2}\\ \end{align*} So, the equations become \begin{align*} \omega_1^2(Q_1 +cQ_2)&=-(1+bl_2c)(\ddot{Q}_1+c\ddot{Q}_2)\\ -\omega^2Q&=\ddot{Q}\\ \end{align*} where $$\omega=\sqrt{\frac{\omega_1^2}{1+bl_2c}}, Q=Q_1+cQ_2$$.