https://web2.ph.utexas.edu/~vadim/Classes/2024f-emt/sep.pdf
Second order differential equation of the form
let the exponential ansatz be
The equation becomes and the characteristic equation becomes
Case 1:
r^2 = - \lambda \\ r = \pm i \sqrt{ \lambda } \end{align}$$ Using Euler's equation $e^{i \theta} = \cos{\theta} + i\sin{\theta}$, the general solution is a linear combination of sine and cosines: $$\boxed{x(t) = A \cos{\lambda t} + B\sin{\lambda t}}$$ *Case 2:* $\lambda < 0$ define a positive constant $\mu = \sqrt{ - \lambda }$ so $\lambda = - \mu^2$ $$\begin{align} r = \pm \mu = \pm \sqrt{ -\lambda } \end{align}$$ General solution: $$\begin{aligned} \text{exponential form: } \boxed{x(t) = C e^{\sqrt{ - \lambda }t} + De^{-\sqrt{ - \lambda }t}} \\ \text{hyperbolic form: } \boxed{x(t) = C \cosh{\sqrt{ - \lambda }t} + D\sinh{\sqrt{ - \lambda }t}} \end{aligned}$$