99 Percentile Qs Bank for JEE MainMathematicsDifferential Equations
The general solution of the differential equation x y x (y d x+x d y)=y y x (x d y-y d x) is
Options
- A(x y)= x y +C
- B( y x )= C x y
- C(x y)= x y +C
- Dx+y+C=0
Correct answer
B. ( y x )= C x y
Step-by-step solution
We have, x ( y x )(y d x+x d y)=y y x (x d y-y d x) x y ( y x ) d y-y^2 ( y x ) d x=x y ( y x ) d x+x^2 ( y x ) d y [x y ( y x )-x^2 ( y x ) ] d y= [x y ( y x )+y^2 ( y x ) ] d x d y d x = x y ( y x )+y^2 ( y x ) x y ( y x )-x^2 ( y x ) d y d x = y x ( y x )+ ( y x )^2 ( y x ) y x ( y x )- ( y x ) Put, y=v x d y d x =v+x d v d x v+x d v d x = v v+v^2 v v v- v x d v d x = 2 v v v v- v v v- v v v d v=2 d x x ( v- 1 v ) d v=2 d x x | v|- |v|=2 |x|+ |C₁ | | v v |- |x|^2= |C₁ | | v v x^2 |= |C₁ | v v x^2 =C₁ ( y x ) y x