MHT CET20244 May 2024Evening ShiftMathematicsDifferential EquationsActual
The general solution of the differential equation 1 x ~d y ~d x = ⁻¹ x is
Options
- Ay+ x^2 ⁻¹ x 2 + c =0 , where c is a constant of integration.
- By+x ⁻¹ x+ c =0 , where c is a constant integration.
- Cy-x- ⁻¹ x+ c =0 , where is a constant of integration.
- Dy= x^2 ⁻¹ x 2 - 1 2 (x- ⁻¹ x )+ c , where c is constant of integration.
Correct answer
D. y= x^2 ⁻¹ x 2 - 1 2 (x- ⁻¹ x )+ c , where c is constant of integration.
Step-by-step solution
aligned & 1 x ~d y ~d x = ⁻¹ x & ~d y=x ⁻¹ x ~d x aligned Integrating on both sides we get, aligned & y= x ⁻¹ x ~d x & y= ⁻¹ x x ~d x- ( d ~d x ⁻¹ x x ~d x ) d x & y= ⁻¹ x x^2 2 - ( 1 1+x^2 x^2 2 ) d x & y= ⁻¹ x x^2 2 - 1 2 ( x^2 1+x^2 ) d x & = x^2 ⁻¹ x 2 - 1 2 ( x^2+1 x^2+1 - 1 1+x^2 ~d x ) & y= x^2 ⁻¹ x 2 - 1 2 (x- ⁻¹ x )+ c aligned