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
$$\begin{gathered} \frac{d^{2}y}{d{x}^2}=2\frac{a}{x^3} \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{2}{x}\left(-\frac{a}{x^2}\right) \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=-\frac{2}{x}\left(\frac{dy}{dx}\right)\left[\text{Using}\left(2\right)\right] \\ \Rightarrow \frac{d^{2}y}{d{x}^2}+\frac{2}{x}\left(\frac{dy}{dx}\right)=0 \end{gathered}$$
Hence, the given function is the solution to the given differential equation.