solve the differential equation

Question

dy=(x+y+1)² dx

Answer 1

The given differential equation is

\(\frac {dy}{dx} = (x + y + 1)^2 \quad ....(i)\)

Let \(x + y + 1 = v\)

⇒ \(\frac{dy}{dx} = \frac {dv}{dx} - 1\)

\(\frac {dv}{dx} - 1 = v^2\)

⇒ \(\frac{dv}{v^2 + 1} = dx\)

Integrating, we get

\(\int \frac{dv}{v^2 + 1} = \int dx\)

\(\tan^{–1} (v) = x + c\)

\( \tan^{–1} (x + y + 1) = x + c\)

which is the required solution.