\( y ^ { \prime } = ( x + y + 1 ) ^ { 2 } \)

We start with the differential equation \[ y' = (x + y + 1)^2. \] **Step 1. Change of Variable** Introduce a new variable \[ w = x + y + 1. \] Differentiate with respect to \( x \): \[ \frac{dw}{dx} = 1 + y'. \] **Step 2. Substitute the Original Equation** Given \( y' = w^2 \), substitute into the derivative of \( w \): \[ \frac{dw}{dx} = 1 + w^2. \] **Step 3. Separate Variables** Separate the variables so that all \( w \)-terms are on one side: \[ \frac{dw}{1 + w^2} = dx. \] **Step 4. Integrate Both Sides** Integrate the left-hand side with respect to \( w \) and the right-hand side with respect to \( x \): \[ \int \frac{dw}{1 + w^2} = \int dx. \] The integral of the left-hand side is \[ \arctan(w) + C_1, \] and the right-hand side integrates to \[ x + C_2. \] We can combine the constants \( C_1 \) and \( C_2 \) into a single constant \( C \): \[ \arctan(w) = x + C. \] **Step 5. Solve for \( w \)** Solve for \( w \) by taking the tangent of both sides: \[ w = \tan(x + C). \] **Step 6. Substitute Back for \( y \)** Recall that \( w = x + y + 1 \), so substitute back: \[ x + y + 1 = \tan(x + C). \] Finally, solve for \( y \): \[ y = \tan(x + C) - x - 1. \] This is the general solution of the differential equation.