Question

The solution of $(x^2+y^2)dx=2xydy$ is
(where $c$ is integration constant)

• A. $|c(x^2-y^2)|=|x|$
• B. $c(x^2+y^2)=|x|$
• C. $c|(x^2-y^2)|=y$
• D. $c(x^2+y^2)=|y|$

Answer 1

The correct option is A $|c(x^2-y^2)|=|x|$

$(x^2+y^2)dx=2xydy\Rightarrow \frac{dy}{dx}=\frac{x^2+y^2}{2xy}\cdots \left(i\right)$
This is homogeneous D.E.

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

Then, equation $\left(i\right)$ becomes :
$v+x\frac{dv}{dx}=\frac{x^2+v^2{x}^2}{2v{x}^2}\Rightarrow x\frac{dv}{dx}=\frac{1+v^2}{2v}-v$
$\Rightarrow \frac{2v}{1-v^2}dv=\int \frac{dx}{x}\Rightarrow -\ln |1-v^2|=\ln |x|+\ln |c|\Rightarrow \frac{1}{|1-v^2|}=|cx|\Rightarrow \frac{x^2}{|x^2-y^2|}=|cx|$
$\Rightarrow |x|=|c(x^2-y^2)|,$ which is the required solution.