Question

Show that the function $y=Ax+\frac{B}{x}$ is a solution of the differential equation $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=0$.

Answer 1

given $y=Ax+\frac{B}{x}...\left(1\right)$

Differntiating on both sides

$\Rightarrow$ $\frac{dy}{dx}=A+B(-\frac{1}{x^2})$

$\frac{dy}{dx}=A-\frac{B}{x^2}....\left(2\right)$

Differentiating $\frac{d^{2}y}{d{x}^2}=\frac{2B}{x^3}$

$B=\frac{x^3}{2}\frac{d^{2}y}{d{x}^2}....\left(3\right)$

Substituting in (2) $\frac{dy}{dx}=A-\frac{x}{2}\frac{d^{2}y}{d{x}^2}$

$A=\frac{dy}{dx}+\frac{x}{2}\frac{d^{2}y}{d{x}^2}....\left(4\right)$

Substitute (3) and (4) in (1)

$y=(\frac{dy}{dx}+\frac{x}{2}\frac{d^{2}y}{d{x}^2})x+\left(\frac{x^3}{2}\frac{d^{2}y}{d{x}^2}\right)\frac{1}{x}$

$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=0$

Hence proved