Question

A differential equation is given as
$x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=4$
The solution of the differential equation in terms of arbitrary constants $C_1$ and $C_2$ is

• A. $y=C_1{x}^2+C_{2}x+4$
• B. $y=\frac{C_1}{x^2}+C_{2}x+2$
• C. $y=C_1{x}^2+C_{2}x+2$
• D. $y=\frac{C_1}{x}+C_{2}x+$4

Answer 1

The correct option is C $y=C_1{x}^2+C_{2}x+2$

$x^2\frac{d^{2}y}{d{x}^2}-2x\frac{dy}{dx}+2y=4$
Let $x=e^z$
$CF:\frac{d^{2}y}{d{x}^2}=\theta (\theta -1$
$\therefore \theta (\theta -1)-2\theta +2=0$
$\therefore {\theta }^2-3\theta +2=0$
$(\theta -1)(\theta -2)=0$
$\theta =1,2$
$\theta =C_1{e}^z+C_1{e}^{2z}$
$y=C_1{x}^2+C_{2}x$
$PI=-\frac{4}{(\theta -1)}+\frac{4}{(\theta -2)}$
$=+4(1-\theta {)}^{-1}-\frac{4}{2}{(1-\frac{\theta }{2})}^{-1}$
$=+4(1+\theta +...)-2(1+\frac{\theta }{2}+...)$
$=+4-2=2$ (Neglecting higher order terms)
General solution is
$y=CF+PI$
$y=C_1{x}^2+C_{2}x+2$