Solve (x+y+1)dy/dx=1

(x + y + 1) dy/dx = 1

Take x as the dependent variable

dx/dy = x + y + 1 → dx/dy − x = y + 1

This is linear in x with P = −1, Q = y + 1.

Integrating factor

IF = e^(∫(−1)dy) = e^(−y)

Solution

x e^(−y) = ∫ (y + 1)e^(−y) dy

By parts: ∫ (y + 1)e^(−y) dy = −(y + 1)e^(−y) + ∫ e^(−y) dy = −(y + 1)e^(−y) − e^(−y) = −(y + 2)e^(−y)

x e^(−y) = −(y + 2)e^(−y) + c

x = −y − 2 + c eʸ

x + y + 2 = c eʸ