Let $\mathcal{L}$ be the linear differential operator

$$\mathcal{L} y=y^{\prime \prime \prime}-y^{\prime \prime}-2 y^{\prime}$$

where $^{\prime}$ denotes differentiation with respect to $x$.

Find the Green's function, $G(x ; \xi)$, for $\mathcal{L}$ satisfying the homogeneous boundary conditions $G(0 ; \xi)=0, G^{\prime}(0 ; \xi)=0, G^{\prime \prime}(0 ; \xi)=0$.

Using the Green's function, solve

$$\mathcal{L} y=e^{x} \Theta(x-1)$$

with boundary conditions $y(0)=1, y^{\prime}(0)=-1, y^{\prime \prime}(0)=0$. Here $\Theta(x)$ is the Heaviside step function having value 0 for $x<0$ and 1 for $x>0$.