Question

Solution of the differential equation $2xy\frac{dy}{dx}=x^2+3y^2$ is :
(where $c$ is integration constant)

• A. $x^3+y^2=c{x}^2$
• B. $\frac{x^2}{2}+\frac{y^3}{2}=y^2+c$
• C. $x^2+y^2=|c{x}^3|$
• D. $|x^2-y^2|=c{x}^3$

Answer 1

The correct option is C $x^2+y^2=|c{x}^3|$

$2xy\frac{dy}{dx}=x^2+3y^2\Rightarrow \frac{dy}{dx}=\frac{x^2+3y^2}{2xy}\cdots \left(i\right)$
This is a homogeneous differential equation.

We put $y=vx$ so that $\frac{dy}{dx}=v+x\frac{dv}{dx}$

$\therefore$ Equation $\left(i\right)$ becomes:
$v+x\frac{dv}{dx}=\frac{x^2+3v^2{x}^2}{2v{x}^2}\Rightarrow x\frac{dv}{dx}=\frac{1+3v^2}{2v}-v\Rightarrow \int \frac{2v}{1+v^2}dv=\int \frac{dx}{x}$
$\Rightarrow \ln (1+v^2)=\ln |x|+\ln |c|\Rightarrow 1+v^2=|cx|\Rightarrow x^2+y^2=|c{x}^3|$