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!
Solve These Irregular Coupled Second Order Differential Equations
ordinary differential equationsphysics
Best Answer
Meant to be a comment
I have never solved coupled differential equations, but here goes my try:
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 $\omega=\sqrt{\frac{\omega_1^2}{1+bl_2c}}, Q=Q_1+cQ_2$.