Question

Consider the differential equaion $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-4y=0$ with the boundary condition of $y\left(0\right)=0$ and $y\left(1\right)=1$. The complete solution of the differential equation is

• A. $x^2$
• B. $sin\left(\frac{\pi x}{2}\right)$
• C. $e^{x}sin\left(\frac{\pi x}{2}\right)$
• D. $e^{-x}sin\left(\frac{\pi x}{2}\right)$

Answer 1

The correct option is A $x^2$

Given DE is
$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-4y=0$
Let $z=logx$
$\therefore$ $\left(D^{\prime }\right(D^{\prime }-1)+D^{\prime }-4)y=0$
Where $D^{\prime }=\frac{d}{dz}$
$(D^2-4)y=0$
Aux.eqn. $m^2-4=0$
$m=-2,2$
The solution is
$y=c_1{e}^{+2x}+c_2{e}^{-2z}$
$y=c_1{x}^2+c_2{x}^{-2}$
Given boundary condition is $y\left(0\right)=0$
$\Rightarrow$ $0=0+\frac{c_2}{0}$
$\Rightarrow$ $c_2=0$
$\therefore$ $y=c_1{x}^2$
From $y\left(1\right)=1$
$a=b$
$\Rightarrow$ $c_1=1$
$\therefore$ $y=x^2$
The complete solution of given differetial equation is $y=x^2$