Chapter 1.5, Problem 31P

a.

Program Plan Intro

To show:That general solution of the differential equation $\frac{dy}{dx}+P\left(x\right)y=0$ is,

  $y_c\left(x\right)=C{e}^{-{\displaystyle \int P\left(x\right)dx}}$

b.

Program Plan Intro

To prove: That particular solution of the differential equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ is,

  $y_p\left(x\right)=e^{-{\displaystyle \int P\left(x\right)dx}}\left[{\displaystyle \int \left(Q\left(x\right)e^{-{\displaystyle \int P\left(x\right)dx}}\right)dx}\right]$

c.

Program Plan Intro

To prove:The general solution of the differential equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ is $y=y_P+y_C$ where $y_c\left(x\right)$ is any general solution of $\frac{dy}{dx}+P\left(x\right)y=0$ and $y_p\left(x\right)$ is any particular solution of the differential equation $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$ .