Question

The general solution of the differential equation $x^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0$ is

• A. $Ax+B{x}^2$ ($A,B$ are constants)
• B. $Ax+Blogx$ ($A,B$ are constants)
• C. $Ax+B{x}^{2}logx$ ($A,B$ are constants)
• D. $Ax+Bxlogx$ ($A,B$ are constants)

Answer 1

The correct option is D $Ax+Bxlogx$ ($A,B$ are constants)

It is Homogeneous Linear $D.E.$
$x^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0$
Let $z=logx$
$\left[D^{\prime }\right(D^{\prime }-1)-D^{\prime }+1]y=0$
$D^{{}^{\prime }2}-2D^{\prime }+1=0$
$D^{\prime }=1,1$
$y=(A+Bz)e^z$
$y=(A+Blogx)x$