MHT CET202523 Apr 2025Evening ShiftMathematicsDifferential EquationsActual
Solution of (2 y-x) d y d x =1 is
Options
- Ax=2(y-1)+c e ^ - y , where c is the constant of integration
- Bx=2(y-1)+ ce ^ -x , where c is the constant of integration
- Cy =2(x-1)+ ce ^ -x , where c is the constant of integration
- Dy =2(x-1)+ ce ^ - y , where c is the constant of integration
Correct answer
A. x=2(y-1)+c e ^ - y , where c is the constant of integration
Step-by-step solution
Given the differential equation (2y-x) dy dx =1 , express it in terms of dx dy : dx dy = 2y - x . Rewriting as a linear differential equation: dx dy + x = 2y . This is of the form dx dy + P(y)x = Q(y) with P(y)=1 and Q(y)=2y . The integrating factor is e^ P(y) dy = e^ 1 dy = e^y . The general solution becomes: x e^y = Q(y) e^y dy + C = 2y e^y dy + C . Using integration by parts where u=2y , dv=e^y dy giving du=2dy , v=e^y : 2y e^y dy = 2ye^y - 2e^y dy = 2ye^y - 2e^y . Substituting: x e^y = 2ye^y - 2e^y + C . Dividi