Question

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

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

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
$\frac{dk}{dx} + k \bigg(- \frac{2}{x}\bigg) = -2x^2$
$I.F, = e^{\int \frac{2}{x}}dx = \frac{1}{x^2}$
$\therefore \, \, \, solution \, \, is \, \, k. \frac{1}{x^2} = \int - 2x^2 . \frac{1}{x^2} dx + \lambda$
$y^2 + 2x^3 = \lambda x^2$
take $\lambda = - c$ (integration constant)