I am looking to find a general solution to ODEs of the form: $$\sum_{i=0}^nf^{(i)}(t)=0$$
for n=2 I obtained the solution: $$y=\exp\left(-\frac{t}{2}\right)\left[A\cos\left(\frac{\sqrt{3}}2t\right)+B\sin\left(\frac{\sqrt{3}}2t\right)\right]$$ And they all follow a general form of: $$y=\sum_{i=1}^nA_ie^{x_it}$$ where $x_i$ are the solutions to the equation: $$\sum_{k=0}^nx^k=0$$ are there any way I can come up with a general formula for $y$ in terms of the respective exponential and trig functions? Thanks