Chapter 8.4, Problem 29ES

(a) Prove that any function $y=y(x)$ defined by Equation (7) will be a solution to (3).

(b) Consider the initial -value problem

$\frac{dy}{dx}+p(x)y=q(x),y(x_0)=y_0$

where the functions $p(x)$ and $q(x)$ are both continuous on some open interval. Using the general solution for a first-order linear equation, prove that this initial-value problem has a unique solution on the interval.