Question

A family of curves has the differential equation $xy\frac{dy}{dx}=2y^2-x^2$. Then the family of curves is:

• A. $y^2=c{x}^2+x^3$
• B. $y^2=c{x}^4+x^3$
• C. $y^2=x+c{x}^4$
• D. $y^2=x^2+c{x}^4$

Answer 1

The correct option is D $y^2=x^2+c{x}^4$

$\Rightarrow y\frac{dy}{dx}=\frac{2y^2}{x}-x$

Let $y^2=t\Rightarrow 2y\frac{dy}{dx}=\frac{dt}{dx}$

$\Rightarrow \frac{1}{2}\frac{dt}{dx}=\frac{2t}{x}-x$

$\Rightarrow \frac{dt}{dx}-4\frac{t}{x}=-2x$

Integrating factor =$e^{\int \frac{-4}{x}dx}=e^{-4lnx}=\frac{1}{x^4}$

Solution $t.\frac{1}{x^4}=\int \frac{1}{x^4}.-2xdx$

$\Rightarrow y^2.\frac{1}{x^4}=\frac{1}{x^2}+c$

$\Rightarrow y^2=x^2+c{x}^4$