Question

The solution of the differential equation $x y^{2} \,d y-\left(x^{3}+y^{3}\right) d x=0$ is

• A. $y^{3}=3 x^{3}+c$
• B. $y^{3}=3 x^{3} \log (c x)$
• C. $y^{3}=3 x^{3}+\log (c x)$
• D. $y^{3}+3 x^{3}=\log (c x)$

Answer 1

Answer: (B)

Given differential equation can be rewritten as
$\frac{d y}{d x}=\frac{x^{3}+y^{3}}{x y^{2}}$
It is a homogeneous differential equation.
Put $y=v x $
$\Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
$\therefore x \frac{d v}{d x}+v=\frac{x^{3}+v^{3} x^{3}}{x^{3} v^{2}}$
$\Rightarrow x \frac{d v}{d x}+v=\frac{1+v^{3}}{v^{2}}$
$\Rightarrow x \frac{d v}{d x}=\frac{1}{v^{2}}$
$\Rightarrow v^{2} d v=\frac{d x}{x}$
On integrating both sides,
we get
$\frac{v^{3}}{3}=\log x+\log c$
$\Rightarrow \frac{1}{3}\left(\frac{y}{x}\right)^{3}=\log x+\log c$
$\Rightarrow y^{3}=3 x^{3} \log c x$