COMEDK2024Evening ShiftMathematicsDifferential EquationsActual
The general solution of the differential equation (1+ y)(d x-d y)+2 x d y=0 is
Options
- Ay( x+ x)= x+c e^ -x
- Bx( y+ y)= y+c e^ -y
- Cx( y+ y)= y+c e^y
- Dy( x+ x)= x+c e^x
Correct answer
B. x( y+ y)= y+c e^ -y
Step-by-step solution
The given differential equation is (1+ y)(dx - dy) + 2x dy = 0 . Rearranging the terms to form a linear differential equation in x : (1+ y) dx = (1+ y - 2x) dy dx dy = 1+ y - 2x 1+ y = 1 - 2x 1+ y dx dy + ( 2 1+ y ) x = 1 This is a linear differential equation of the form dx dy + P(y)x = Q(y) , where P(y) = 2 1+ y = 2 y y + y . The integrating factor (IF) is e^ P(y) dy = e^ 2 y y + y dy . Let I = 2 y y + y dy . Note that 2 y = ( y + y) + ( y - y) . I = (1 + y - y y + y ) dy = y + | y + y| . Thus, IF = e^ y + | y +