Consider the (one-dimensional) linear operator \mathcal{L}=\frac{d^{2}}{d x^{2}}-a^{2} where a is a positive constant. ) Show that, for r > 0 and the boundary conditions y(xy'(x = 0) = 0, the

Green's function of this linear operator is0) = 0 and G\left(x, x^{\prime}\right)=\left\{\begin{array}{ll} 0, & \text { for } xx^{\prime} \end{array}\right. and determine the function f(x – x').* [5] Use the Green's function from part (a) to obtain the solution y(x) for x > 0to the in homogeneous differential equation y"(x) – a²y(x) = e¯ with the boundary conditions y(0) = 0 and y'(0) = 0. [5]