While solving Euler's equation of motion for an asymmetric top, I cam across the three following equations :
$$\dot{\omega_1}+\omega_2 \omega_3=0$$ $$\dot{\omega_2}-\omega_1 \omega_3=0$$ $$\dot{\omega_3}+\frac{\omega_2 \omega_1}{\mu^2}=0$$
The only information that I have been provided is that at time $t=0$, $\omega=N\mu \hat{i}+N\hat{k}$. So I know $\omega_1, \omega_2 $ and $\omega_3$ at time $0$. How can I decouple and solve the above set of equations ?
I've already been given the solution as follows :
$$\omega_1 =\frac{N\mu}{\cosh(Nt)}$$ $$\omega_2 ={N\mu}\tanh(Nt)$$ $$\omega_3 =\frac{N}{\cosh(Nt)}$$
How do I solve the equations myself ? I've solved linear coupled differential equations using matrices, however this one seems to be nonlinear because of the $\omega_i\omega_j$ terms.