Question

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

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

Answer 1

Answer: (B)

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