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
\[\frac{d^2 y}{d x^2} = A B^2 e^{Bx} \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{\left( AB e^{Bx} \right)^2}{\left( A e^{Bx} \right)}\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{1}{y} \left( \frac{dy}{dx} \right)^2 ...........\left[\text{Using }\left( 1 \right)\text{ and }\left( 2 \right) \right]\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{1}{y} \left( \frac{dy}{dx} \right)^2\]

Hence, the given function is the solution to the given differential equation.