Chapter 2.4, Problem 23P

Show that $\varphi \left(t\right)=e^{2t}$ is a solution of $y\text{'}-2y=0$ and that $y=c\varphi \left(t\right)$ is also a solution of this equation for any value of the constant $c$.

Show that $\varphi \left(t\right)=1/t$ is a solution of $y\text{'}+y^2=0$ for $t>0$ but that $y=c\varphi \left(t\right)$ is not a solution of this equation unless $c=0$ or $c=1$. Note that the equation of part (b) is nonlinear, whereas that of part (a) is linear.