The general solution of the differential equation (x+y) dy dx = 1 is

Step-by-step solution

Given differential equation is (x+y) dy dx = 1 Rewriting the equation, we get dx dy = x + y dx dy - x = y This is a linear differential equation of the form dx dy + Px = Q , where P = -1 and Q = y . Integrating Factor (IF) = e^ P dy = e^ -1 dy = e^ -y The general solution is given by x ( IF ) = Q ( IF ) dy + c x e^ -y = y e^ -y dy + c Using integration by parts, x e^ -y = y(-e^ -y ) - 1 (-e^ -y ) dy + c x e^ -y = -y e^ -y - e^ -y + c Multiplying both sides by e^y , we get x = -y - 1 + c e^y x + y + 1 = c e^y Answer