Basis for Solution Space of Differential Equation

Question

Quick question for my differential equations and linear algebra homework. Say we have a differential equation that has the general solution $$ y = c_1e^{-2x} + c_2e^{-2x} $$ where $c1$, $c2$ are arbitrary constants.

I need to find the basis for this. If the exponents had opposite signs, the basis would just be: $$ \{e^{-2x},\ e^{2x}\} $$ But since they are the same, and the exponent has an algebraic multiplicity of $2$, I'm not quite sure of the proper way to state the basis. I could see the basis being

1. $\{e^{-2x}\}$, or
2. $\{e^{-2x},\ e^{-2x}\}$, or even
3. $\{0,\ e^{-2x}\}$.

Thanks in advance guys.

Answer 1

HINT:

Notice, if the roots are equal then the general solution of differential equation: $\frac{d^2y}{dx^2}+4x\frac{dy}{dx}+4x^2y=0$ is given as $$y=(c_1+xc_2)e^{-2x}$$

while the basis, $e^{-2x}$ & $e^{2x}$ shows that roots are distinct of differential equation $\frac{d^2y}{dx^2}-4x^2y=0$ whose general solution is given as $$y=c_1e^{2x}+c_2e^{-2x}$$