AP EAMCET202418 May 2024Morning ShiftMathematicsDifferential EquationsActual
The general solution of the differential equation (x-y-1) d y=(x+y+1) d x is
Options
- A⁻¹ ( y+1 x )- 1 2 (x^2+y^2+2 y+1 )=c
- B(x-y)+ (x+y)=c
- Cy^2-x^2+x y-3 y-x=c
- D(x-y-1)^2(x+y+1)^3=c
Correct answer
A. ⁻¹ ( y+1 x )- 1 2 (x^2+y^2+2 y+1 )=c
Step-by-step solution
Given differential equation d y d x = x+y+1 x-y-1 aligned & Let x =X+h, y=Y+k d y d x = d Y d X & d Y d X = X+Y+h+k+1 X-Y+h-k-1 aligned Since, h+k+1=0h-k-1=0 After solving this we get h=0, k=-1 So, d Y d X = X+Y X-Y (X-Y) d Y=(X+Y) d X X d Y-Y d X=X d X+Y d Y X d Y-Y d X X^2 (1+ Y^2 X^2 ) = X d X+Y d Y X^2+Y^2 aligned & d ⁻¹ ( Y X )= 1 2 d ( (X^2+Y^2 ) ) & ⁻¹ ( Y X )= 1 2 (X^2+Y^2 )+C aligned aligned & ⁻¹ ( y+1 x )= 1 2 (x^2+(y+1)^2 )+ C & ⁻¹ ( y+1 x )- 1 2 (x^2+y^2+2 y+1 )= C aligned