Question

Verify that y = $a{e}^{-x}$ is a solution of
$\frac{d^{2}y}{d{x}^2}=\frac{1}{y}{\left(\frac{dy}{dx}\right)}^2$

Answer 1

Given, $y=a{e}^{-x}$

Now, $\frac{dy}{dx}=-a{e}^{-x};\frac{d^{2}y}{d{x}^2}=a{e}^{-x}$

Now, $\frac{d^{2}y}{d{x}^2}=a{e}^{-x}$ & $\frac{1}{y}{\left(\frac{dy}{dx}\right)}^2=\frac{1}{a{e}^{-x}},a^2{e}^{-2x}$

$=a{e}^{-x}$

so, $\frac{d^{2}y}{d{x}^2}=\frac{1}{y}{\left(\frac{dy}{dx}\right)}^2$.

So, $y=a{e}^{-x}$ is solution of that diff. eqn.