MHT CET202523 Apr 2025Morning ShiftMathematicsDifferential EquationsActual
If x dy d x = y ( y - x+1) , then the solution of the equation is
Options
- Ax y = cy , where c is the constant of integration
- By x = cy , where c is the constant of integration
- Cx y = c x , where c is the constant of integration
- Dy x = c x , where c is the constant of integration
Correct answer
D. y x = c x , where c is the constant of integration
Step-by-step solution
The differential equation x dy dx = y( y - x + 1) is homogeneous. Dividing both sides by x gives dy dx = y x ( (y/x) + 1) , showing that the equation depends on the ratio y/x . Substitute y = vx so that dy dx = v + x dv dx . Substituting into the equation yields v + x dv dx = v( v + 1) , which simplifies to x dv dx = v v . Separate the variables to dv v v = dx x . Integrate both sides: the left side becomes du u after substituting u = v , giving | v| , while the right side is |x| + C . Equating gives | v| = |x| + C