Step 1: Understanding the Question:
The given differential equation is of the form $\frac{dy}{dx} = f(ax + by + c)$. We use substitution to solve it. Step 3: Detailed Explanation:
Let $v = x + y + 2$.
Differentiating with respect to $x$: $\frac{dv}{dx} = 1 + \frac{dy}{dx}$.
From the original equation: $\frac{dy}{dx} = \frac{1}{x + y + 1} = \frac{1}{v - 1}$.
Substitute $\frac{dy}{dx}$ in the $v$ derivative:
$\frac{dv}{dx} = 1 + \frac{1}{v - 1} = \frac{v - 1 + 1}{v - 1} = \frac{v}{v - 1}$.
Separate the variables:
$\frac{v - 1}{v} dv = dx \implies (1 - \frac{1}{v}) dv = dx$.
Integrating both sides:
$\int (1 - \frac{1}{v}) dv = \int dx$
$v - \log|v| = x + c$
Substitute $v = x + y + 2$:
$(x + y + 2) - \log|x + y + 2| = x + c$
$y + 2 - c = \log|x + y + 2|$
Let $c' = 2 - c$ be another constant:
$y = \log(x + y + 2) + C$ Step 4: Final Answer:
The solution is $y = \log(x + y + 2) + c$.