Question

Show that y = Ae^Bx is a solution of the differential equation $\frac{d^{2}y}{d{x}^2}=\frac{1}{y}{\left(\frac{dy}{dx}\right)}^2$.

Answer 1

We have,
$y=A{e}^{Bx}$ ...(1)
Differentiating both sides of (1) with respect to $x$, we get
$\frac{dy}{dx}=AB{e}^{Bx}$ ...(2)
Differentiating both sides of (2) with respect to $x$, we get
$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=A{B}^2{e}^{Bx} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{{\left(AB{e}^{Bx}\right)}^2}{\left(A{e}^{Bx}\right)} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{1}{y}{\left(\frac{dy}{dx}\right)}^2\left[\text{Using}\left(1\right)\text{and}\left(2\right)\right] \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{1}{y}{\left(\frac{dy}{dx}\right)}^2 \end{gathered}$$
Hence, the given function is the solution to the given differential equation.