Chapter 9, Problem 4AAE

(a)

To determine

To Verify: The function $y\left(x\right)=y_1\left(x\right)-y_2\left(x\right)$ satisfies the initial value problem $y^{\prime }+P\left(x\right)y=0,y\left(x_0\right)=0$.

(b)

To determine

To show: If the integrating factor is $v\left(x\right)=e^{{\displaystyle \int P\left(x\right)dx}}$ then $\frac{d}{dx}\left(v\left(x\right)\left[y_1\left(x\right)-y_2\left(x\right)\right]\right)=0$ and also to show that $v\left(x\right)\left[y_1\left(x\right)-y_2\left(x\right)\right]$ is a constant.

(c)

To determine

To show: That $y_1\left(x\right)-y_2\left(x\right)\equiv 0$ on the interval $\left(a,b\right)$ and also $y_1\left(x\right)=y_2\left(x\right)$ for all $a<x<b$.