MHT CET202526 Apr 2025Evening ShiftMathematicsDifferential EquationsActual
The solution of the differential equation. (1+x) dy d x -x y =1-x is
Options
- Ay (1+x)=x+ ce ^x , where c is the constant of integration
- By (1+x)= ce ^x , where c is the constant of integration
- Cy (1-x)=x- ce ^x , where c is the constant of integration
- Dy (1+x)=x~ ce ^ -x , where c is the constant of integration
Correct answer
A. y (1+x)=x+ ce ^x , where c is the constant of integration
Step-by-step solution
Convert the differential equation to standard form by dividing through by 1+x : d y d x - x 1+x y = 1-x 1+x Identify P(x) = - x 1+x and compute the integrating factor: P(x) , d x = (-1 + 1 1+x ) , d x = -x + |1+x| IF = e^ -x + |1+x| = (1+x)e^ -x Multiply through by the integrating factor and integrate: y(1+x)e^ -x = (1-x)e^ -x , d x + C Evaluate the integral using integration by parts with u = 1-x , d v = e^ -x , d x : (1-x)e^ -x , d x = (1-x)(-e^ -x ) - (-e^ -x )(- d x) = -(1-x)e^ -x - e^ -x = xe^ -x Substitute ba