Problem 1: As we saw in class, a linear, second order, homogeneous differential equation has the form a_{2}(x) \frac{d^{2} y}{d x^{2}}+a_{1}(x) \frac{d y}{d x}+a_{0}(x) y=0 where y is a function

of x that is at least twice differentiable and a2(x), a1(x), and ao(x) are functions of x. We will define a function L which takes as inputs (at least) twice differentiable functions y as follows: 1. Choose functions a2(x), a1(x), and ao(x). 2. Write L=a_{2}(x) \frac{d^{2}}{d x}+a_{1}(x) \frac{d}{d x}+a_{0}(x) 3. Define L(y)as L(y)=\left(a_{2}(x) \frac{d^{2}}{d x}+a_{1}(x) \frac{d}{d x}+a_{0}(x)\right) y =a_{2}(x) \frac{d^{2} y}{d x^{2}}+a_{1}(x) \frac{d y}{d x}+a_{0}(x) y