COMEDK202510 May 2025Morning ShiftMathematicsDifferential EquationsActual
The general solution of the differential equation (x-y) d y=(x+y) d x is
Options
- A⁻¹ ( y x )=x^2+y^2+c
- B⁻¹ ( y x )=c x^2+y^2
- Ce^ ⁻¹ ( y x ) =c x^2+y^2
- De^ ⁻¹ ( y x ) = c x^2+y^2 x
Correct answer
C. e^ ⁻¹ ( y x ) =c x^2+y^2
Step-by-step solution
The given differential equation is (x-y) dy = (x+y) dx , which can be written as dy dx = x+y x-y . This is a homogeneous differential equation. Let y = vx , then dy dx = v + x dv dx . Substituting these into the equation, we get v + x dv dx = x+vx x-vx = 1+v 1-v . x dv dx = 1+v 1-v - v = 1+v-v+v^2 1-v = 1+v^2 1-v . Separating the variables, we have 1-v 1+v^2 dv = dx x . Integrating both sides, 1 1+v^2 dv - v 1+v^2 dv = dx x . ⁻¹(v) - 1 2 (1+v^2) = |x| + C₁ . Substituting v = y x , we get ⁻¹ ( y x ) - 1 2 (1 + y^2 x