The solution of the differential equation x(d y/d x)=yln ((y2/x2)) is (where, c is an arbitrary constant)

Question

The solution of the differential equation x(d y/d x)=yln ((y2/x2)) is (where, c is an arbitrary constant) (A) y=x.ec x + 1 (B) y=x.ec x - 1 (C) y=x2.ec x + 1 (D) y=x.ec x2 + (1/2)

Tardigrade Admin · Accepted Answer

Putting y=xv and (d y/d x)=v+x(d v/d x) So, v+x(d v/d x)=2vln v ⇒ displaystyle ∫ (d x/x)= displaystyle ∫ (d v/v (. 2 ln v - 1 .)) On integrating, we get, y=x⋅ ec x2 + (1/2)

Q. The solution of the differential equation $x \frac{d y}{d x} = y l n (\frac{y ^{2}}{x ^{2}})$ is (where, $c$ is an arbitrary constant)

$y = x . e^{c x + 1}$

$y = x . e^{c x - 1}$

$y = x^{2} . e^{c x + 1}$

$y = x . e^{c x^{2} + \frac{1}{2}}$

Putting $y = xv$ and $\frac{d y}{d x} = v + x \frac{d v}{d x}$
So, $v + x \frac{d v}{d x} = 2 v l n v$
$\Rightarrow \int \frac{d x}{x} = \int \frac{d v}{v ( 2 l n v - 1 )}$
On integrating, we get,
$y = x \cdot e^{c x^{2} + \frac{1}{2}}$

Q. The solution of the differential equation xdxdy​=yln(x2y2​) is (where, c is an arbitrary constant)

y=x.ecx+1

y=x.ecx−1

y=x2.ecx+1

y=x.ecx2+21​