Given the following ordinary differential equation (ODE)
$$\frac{dy}{dx} = a y$$
its general solution is $y = c e^{a x}$, where $c$ is a constant. If we know
$$y = c_{1} e^{a_{1} x} + c_{2} e^{a_{2} x}$$
what is the corresponding ordinary differential equation for this solution?
Best Answer
General method: From an expression with two parameters $c_1, c_2$, to find the second-order DE with that solution.
Step 1, solve for $c_1$
Step 2, differentiate to eliminate $c_1$
Step 3, solve for $c_2$
Step 4, differentiate to eliminate $c_2$.
The result is an equation involving $y, y', y''$.