MHT CET202520 Apr 2025Morning ShiftMathematicsDifferential EquationsActual
The general solution of x(x-1) dy d x =x^3(2 x-1)+(x-2) y is
Options
- Ay(x-1)=x^3+c(x-1) , where c is the constant of integration.
- By =x^3(x-1)+ c , where c is the constant of integration.
- Cy(x-1)=x^3(x-1)+c x^2 , where c is the constant of integration.
- Dy(x-1)=x^3(x-1)+c, where c is the constant of integration.
Correct answer
C. y(x-1)=x^3(x-1)+c x^2 , where c is the constant of integration.
Step-by-step solution
The differential equation is x(x-1) dy dx =x^3(2x-1)+(x-2)y . Rewriting in standard linear form dy dx +P(x)y=Q(x) after division by x(x-1) yields dy dx - x-2 x(x-1) y= x^2(2x-1) x-1 . Using partial fractions, x-2 x(x-1) = 2 x - 1 x-1 , so P(x)= 1 x-1 - 2 x . The integrating factor is e^ P(x)dx =e^ | x-1 x^2 | = x-1 x^2 . The general solution becomes y x-1 x^2 = (2x-1)dx+C , where the right side simplifies to x^2-x+C . Multiplying both sides by x^2 gives y(x-1)=x^2(x^2-x+C)=x^4-x^3+Cx^2=x^3(x-1)+Cx^2 . This matches