Chapter 2.3, Problem 28E

Constant Multiples of Solutions.

a. Show that $y=e^{-x}$ is a solution of the linear equation

$dy/dx+y=0$, … $\left(16\right)$

and $y=x^{-1}$ is a solution of the nonlinear equation

$dy/dx+y^2=0$. … $\left(17\right)$

b. Show that for any constant $C$, the function $C{e}^{-x}$ is a solution of equation $\left(16\right)$, while $C{x}^{-1}$ is a solution of equation $\left(17\right)$ only when $C=0$ or $1$.

c. Show that for any linear equation of the form

$\frac{dy}{dx}+P\left(x\right)y=0$,

if $\hat{y}\left(x\right)$ is a solution, then for any constant $C$ the function $C\hat{y}\left(x\right)$ is also a solution.