Answer
${e^{-3x}, e^x}$ is a basis for the solution space.
Work Step by Step
Solve the characteristic equation for the differential equation. $$r^2+2r-3=0$$
Factor and solve for the roots. $$(r-1)(r+3)=0$$ $$r=2,-1$$
The general equation is equal to $y=C_1e^{-3x}+C_2e^{x}$
Therefore, ${e^{-3x}, e^x}$ is a basis for the solution space.