Solve the differential equation : ` (x + y) (dy)/(dx) = 1` .

1.	Solve the differential equation : ` (x + y) (dy)/(dx) = 1` .
Answer» Given , ` (x + y) (dy)/(dx) = 1` ` therefore (dy)/(dx) = x + y` `rArr (dx)/(dy) - x = y ` This is linear differential equation with x as dependent veriable of the from ` (dx)/(dy) + P x = Q . ` ` therefore P = - 1, Q = y ` The integrating factor , I.F. ` = e^(int" P dy" )= e^(int-1 dy) = e^(-y)` ` therefore ` The solution is , ` x (I.F.) = int Q (I.F.) dy + C ` ` rArr xe^(-y) = intye^(-y) dy + C ` ` rArr xe^(-y) = y [-e^(-y)] - int 1 (-e^(-y)) dy + C ` `rArr ex^(-y) = - ye^(-y) - e^(-y) + C ` ` rArr e^(-y) (x + y + 1) = C ` ` rArr x + y + 1 = Ce^(y)` is general solution .

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