I can now recognise the order and the type of differential equations.Let's say $\frac{dy}{dx} = x^2 - 1$ is a first order ODE, $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 + y = 0$ is a second order ODE and so on. I am having trouble to obtain a differential equation from a given function. I could find the differential equation for $y = e^x(A \cos x + B \sin x)$ and the steps that I followed are as follows.
$ \begin{align*} \frac{dy}{dx} &= e^x(A \cos x + B \sin x) + e^x(-A \sin x + B \cos x) \\ &= y +e^x(-A \sin x + B \cos x) \tag{1} \\ \frac{d^2 y}{dx^2} &= \frac{dy}{dx} + e^x(-A \sin x + B \cos x) + e^x(-A \cos x - B \sin x) \\ &= \frac{dy}{dx}+\left(\frac{dy}{dx} - y\right) - y \end{align*} $ using the orginal function and $(1)$. Finally, $ \frac{d^2y}{dx^2} - 2 \frac{dy}{dx} + 2y = 0 ,$ which is the required differential equation.
Similarly, if the function is $y=(A\cos2t + B\sin2t)$, the differential equation that I get is $ \frac{d^2y}{dx^2} + 4y = 0 $ following similar steps as above.
My question is how do I obtain the differential equations for the following functions using similer procedures
$y = Ae^{3x} + Be^{2x}$
$xy = Ae^{x} + Be^{-x} + x^{2}$
I am not looking for a solution in determinant form using vector spaces or any other linear algebra/matrices.Please provide step by step solution for the function in order to obtain a particular differential equation.
Thank you in advance.