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Question
Find the general solution of the differential equation \[\frac{dy}{dx} = \frac{x + 1}{2 - y}, y \neq 2\]
Solution
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
\[\int\left( 2 - y \right)dy = \int\left( x + 1 \right)dx\]
\[ \Rightarrow 2y - \frac{y^2}{2} = \frac{x^2}{2} + x + C_1 \]
\[ \Rightarrow \frac{x^2}{2} + x + C_1 - 2y + \frac{y^2}{2} = 0\]
\[ \Rightarrow x^2 + 2x + y^2 + 2 C_1 - 4y = 0\]
\[ \Rightarrow x^2 + y^2 + 2x - 4y + C = 0 ........\left[\text{Where, }C = 2 C_1 \right]\]
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