Chapter 1.1, Problem 48P

(a)

Program Plan Intro

To show: $y\left(x\right)=C{x}^4$ is the solution of the differential equation $x{y}^{\prime }=4y$ .

(b)

Program Plan Intro

To show: Function $y\left(x\right)=\{\begin{array}{cc}-x^4& \text{if }x<0,\\ x^4& \text{if }x\ge 0\end{array}$ is the solution of $x{y}^{\prime }=4y$ for all $x$ .

(c)

Program Plan Intro

To verify: Any given point $\left(a,b\right)$ in the plane, the differential equation $x{y}^{\prime }=4y$ has exactly one solution $y\left(x\right)$ satisfying the condition $y\left(a\right)=b$ .