MHT CET202613 April 2026Morning ShiftMathematicsDifferential EquationsActual
The solution of the differential equation dy dx = x+y x-y is
Options
- Ac(x^2 + y^2)^ 1 2 + e^ ⁻¹ ( y^2 x ) = 0 , where c is an arbitrary constant
- Bc(x^2 + y^2)^ 1 2 = e^ ⁻¹ ( y x ) , where c is an arbitrary constant
- Cc(x^2 - y^2) = e^ ⁻¹ ( y x ) , where c is an arbitrary constant
- Dc(x^2 + y^2) = e^ ⁻¹ ( y x ) , where c is an arbitrary constant
Correct answer
B. c(x^2 + y^2)^ 1 2 = e^ ⁻¹ ( y x ) , where c is an arbitrary constant
Step-by-step solution
The given differential equation is dy dx = x+y x-y . Substituting y = vx , we get dy dx = v + x dv dx . 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 get: 1-v 1+v^2 dv = dx x Integrating both sides: 1 1+v^2 dv - v 1+v^2 dv = dx x ⁻¹v - 1 2 (1+v^2) = |x| + C Substituting v = y x : ⁻¹ ( y x ) - 1 2 (1+ y^2 x^2 ) = |x| + C ⁻¹ ( y x ) - 1 2 ( x^2+y^2 x^2 ) = |x| + C ⁻¹ ( y x ) - 1 2 (x^2+y^2) + |x| = |x| + C ⁻¹ ( y x ) - 1 2 (x^2+y^2) = C (x^2+y^2)^ 1 2