Consider the nonhomogeneous initial-value problem $ L[y] := y'''- 2y'' - y' + 2y = g(x); x \in R$ $y(x_0) = 0; y'(x_0) = 0; y''(x_0) = 0$
Show that $y_1(x) = e^{2x}, y_2(x) = e^x$ and $y_3(x) = e^{-x}$ form a fundamental set of solutions of the homogeneous differential equation $ L[y] = 0 $
I am confused on how to do this is as there is a $y'''$. Do I just substitute $y_1,y_2$ and $y_3$ into $L[y]$ and show the wronskian is not 0.
You can do what you suggest: show each is a solution, show they are linearly independent using the Wronskian, and then argue using a general theorem that an $n$th order linear homogeneous ODE has $n$ linearly independent solutions. Alternatively, you can use the same theorem and solve the characteristic equation
$$\lambda^3 - 2\lambda^2 - \lambda + 2 = 0$$