Question

The solution of the differential equation $x\frac{d y}{d x}=yln \left(\frac{y^{2}}{x^{2}}\right)$ is (where, $c$ is an arbitrary constant)

• A. $y=x.e^{c x + 1}$
• B. $y=x.e^{c x - 1}$
• C. $y=x^{2}.e^{c x + 1}$
• D. $y=x.e^{c x^{2} + \, \frac{1}{2}}$

Answer 1

Answer: (D)

Putting $y=xv \, $ and $ \, \frac{d y}{d x}=v+x\frac{d v}{d x}$
So, $v+x\frac{d v}{d x}=2vln v$
$\Rightarrow \, \displaystyle \int \frac{d x}{x}=\displaystyle \int \frac{d v}{v \left(\right. 2 ln v - 1 \left.\right)}$
On integrating, we get,
$ \, y=x\cdot e^{c x^{2} + \, \frac{1}{2}}$