Question

Verify that y = \[\frac{a}{x} + b\] is a solution of the differential equation
\[\frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]

Answer 1

We have, \[y = \frac{a}{x} + b   ..............(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = - \frac{a}{x^2}   ..............(2)\]

Differentiating both sides of (2) with respect to x, we get
\[\frac{d^2 y}{d x^2} = 2\frac{a}{x^3}\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - \frac{2}{x}\left( - \frac{a}{x^2} \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - \frac{2}{x}\left( \frac{dy}{dx} \right) ...........\left[\text{Using }\left( 2 \right) \right]\]
\[ \Rightarrow \frac{d^2 y}{d x^2} + \frac{2}{x}\left( \frac{dy}{dx} \right) = 0\]

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