Underdamped free vibration proof

Question

I need to prove the solution form of:

$$y''+2cwy'+wy=0$$

My book says, after assuming a solution of the form $Ce^pt$, you can show that:

$$y=[A\sin(wt)+B\sin(sw)] \cdot e^pt$$

I tried using the characteristic equation on the original ODE but it didn't get me anywhere. Any ideas? Obviously, just plugging in the suggested solution isn't a sufficient proof...Any ideas?

Answer 1

Obviously, just plugging in the suggested solution isn't a sufficient proof

If for any initial conditions there is a solution of the suggested form, that is a proof, because of the uniqueness of solutions to ODE. Any differential equation $y'' = f(y,y')$ satisfies the uniqueness theorem for [Lipschitz] continuous $f$.

Answer 2

Your question should have said $y''+2cwy'+w^2y=0$. Your final solution is also incorrect. You should have persevered with the characteristic equation. Always remember that the general solution has complex coefficients. If you do not allow complex numbers some differential equations will not be solvable via that method. In this case when $(c^2-1)w < 0$ you get complex coefficients. Push ahead and you will be able to factorize out the exponential term, and you can prove that the remaining term is only real when it is sinusoidal.