[Math] Find the differential equation given the general solution

Question

Given that $x(t)=(c_1+c_2 t + c_3 t^2)e^t$ is the general solution to a differential equation, how do you work backwards to find the differential equation?

Answer 1

Say $f(t) = c_1 + c_2t + c_3t^2$
the given general solution is $$x(t) = fe^t$$ Since you have $3$ arbitrary constants, the required DE must be of order $3$. So you need to differentiate exactly 3 times : $$\begin{align} x' &= (f'+f)e^t \\ x''&=(f''+2f'+f)e^t \\x'''&=(f''' + 3f''+3f'+f)e^t\end{align}$$

Its trivial to eyeball the required DE : $x'''-3x''+3x'-x=f'''e^t = 0$