Question

The general solution of the differential equation ($y^2-x^3)dx-xydy=0(x\ne 0)$ is : (where $c$ is a constant of integration)

• A. $y^2+2x^3+c{x}^2=0$
• B. $y^2+2x^2+c{x}^3=0$
• C. $y^2-2x^3+c{x}^2=0$
• D. $y^2-2x^2+c{x}^3=0$

Answer 1

Answer: (A)

xy $\frac{dy}{dx}-y^2+x^2=0$
put $y^2=k\Rightarrow y\frac{dy}{dx}=\frac{1}{2}\frac{dk}{dx}$
$\therefore$ given differential equation becomes
$\frac{dk}{dx}+k(-\frac{2}{x})=-2x^2$
$I.F,=e^{\int \frac{2}{x}}dx=\frac{1}{x^2}$
$\therefore solutionisk.\frac{1}{x^2}=\int -2x^2.\frac{1}{x^2}dx+\lambda$
$y^2+2x^3=\lambda x^2$
take $\lambda =-c$ (integration constant)