Chapter 16, Problem 3AAE

(a)

To determine

To show: The function satisfying the equation $v\left(x\right)y={\displaystyle \int v\left(x\right)Q\left(x\right)dx+C}$ for $v\left(x\right)=e^{{\displaystyle \int P\left(x\right)dx}}$ is a solution to the to the first-order linear equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$.

(b)

To determine

To show: If $C=y_{0}v\left(x_0\right)-{\displaystyle \underset{x_0}{\overset{x}{\int }}v\left(t\right)Q\left(t\right)dt}$ then any solution y in part (a) satisfies the initial condition $y\left(x_0\right)=y_0$.