MHT CET202525 Apr 2025Evening ShiftMathematicsDifferential EquationsActual
The general solution of differential equation (y^2-x^2 ) d x=x y dy (x 0) is
Options
- A2 x^2 x+ y ^2+2 c x^2=0 , where c is the constant of integration
- B2 x^2 x- y ^2+2 c x^2=0 , where c is the constant of integration
- Cx^2 x+ y ^2+2 c x^2=0 , where c is the constant of integration
- Dx^2 x- y ^2+2 c x^2=0 , where c is the constant of integration
Correct answer
A. 2 x^2 x+ y ^2+2 c x^2=0 , where c is the constant of integration
Step-by-step solution
Given the homogeneous differential equation (y^2-x^2 ) dx = x y dy , express dy dx as y^2-x^2 xy . The substitution y = vx with dy dx = v + x dv dx simplifies to v + x dv dx = v^2-1 v . Rearranging gives x dv dx = -1 v , so v dv = - 1 x dx . Integration yields v^2 2 = - |x| + C' . Substituting back v = y x , we obtain y^2 2x^2 = - |x| + C' . Multiplying through by 2x^2 and rearranging, y^2 + 2x^2 x - 2C'x^2 = 0 . Letting 2C' = -2c gives y^2 + 2x^2 x + 2cx^2 = 0 , which corresponds to option A. Final answer: A