Question

Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]

Answer 1

We have,

\[\frac{dy}{dx} = \frac{x + 1}{2 - y}\]

\[ \Rightarrow \left( 2 - y \right)dy = \left( x + 1 \right)dx\]

Integrating both sides, we get

\[\int\left( 2 - y \right)dy = \int\left( x + 1 \right)dx\]

\[ \Rightarrow 2y - \frac{y^2}{2} = \frac{x^2}{2} + x + C_1 \]

\[ \Rightarrow \frac{x^2}{2} + x + C_1 - 2y + \frac{y^2}{2} = 0\]

\[ \Rightarrow x^2 + 2x + y^2 + 2 C_1 - 4y = 0\]

\[ \Rightarrow x^2 + y^2 + 2x - 4y + C = 0 ........\left[\text{Where, }C = 2 C_1 \right]\]