Question

Consider the differential equation $x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=0$. Which of the following is a solution to this differential equation for $x>0$ ?

• A. $e^x$
• B. $x^2$
• C. $1/x$
• D. lnx

Answer 1

The correct option is C $1/x$

It is Homogeneous Linear D.E. of $2^{nd}$ order
$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-y=0$
Let $z=logx$
A.E. is $D^{\prime }(D^{\prime }-1)+D^{\prime }-1=0$
$D^{\prime 2}-1=0$
$D^{\prime }=\pm 1$
$y=C_1{e}^z+C_2{e}^{-z}$
$y=C_{1}x+\frac{C_2}{x}$
$y=x,y=\frac{1}{x}$ are the solution of the given differential equation.