MHT CET202122 Sep 2021Evening ShiftMathematicsDifferential EquationsActual
The general solution of d y d x = x+y x-y is
Options
- A⁻¹ x y + 1 2 |x^2+y^2 |=c
- B⁻¹ y x + 1 2 | x ^2+ y ^2 |= c
- C⁻¹ y x - 1 2 |x^2+y^2 |=c
- D⁻¹ x y - 1 2 |x^2+y^2 |=c
Correct answer
C. ⁻¹ y x - 1 2 |x^2+y^2 |=c
Step-by-step solution
aligned & d y d x = x+y x-y & Put y=v x & d y d x =v+x d v d x & v+x d v d x = x+v x x-v x = 1+v 1-v & x d v d x = 1+v 1-v -v= 1+v-v+v^2 1-v = 1+v^2 1-v & (1-v) 1+v^2 d v= d x x & 1 1+v^2 d v- 1 2 2 v 1+v^2 d v= d x x & ⁻¹(v)- 1 2 |1+v^2 |= |x|+c & ⁻¹ ( y x )- 1 2 |1+ y^2 x^2 |- 1 2 |x^2 |=c aligned aligned & ⁻¹ ( y x )- 1 2 [ | x ^2+ y ^2 x ^2 |+ | x ^2 | ]= c & ⁻¹ ( y x )- 1 2 | x ^2+ y ^2 |= c aligned