Find the general solution of the differential equation $\frac{dy}{dx} = \frac{x+1}{2-y}, (y \neq 2)$.
(N/A) Given the differential equation:
$\frac{dy}{dx} = \frac{x+1}{2-y}$
Separating the variables,we get:
$(2-y) dy = (x+1) dx$
Integrating both sides:
$\int (2-y) dy = \int (x+1) dx$
Performing the integration:
$2y - \frac{y^2}{2} = \frac{x^2}{2} + x + C_1$
Multiplying the entire equation by $2$:
$4y - y^2 = x^2 + 2x + 2C_1$
Rearranging the terms to form the general solution:
$x^2 + y^2 + 2x - 4y + C = 0$,where $C = 2C_1$.
$\frac{dy}{dx} = \frac{x+1}{2-y}$
Separating the variables,we get:
$(2-y) dy = (x+1) dx$
Integrating both sides:
$\int (2-y) dy = \int (x+1) dx$
Performing the integration:
$2y - \frac{y^2}{2} = \frac{x^2}{2} + x + C_1$
Multiplying the entire equation by $2$:
$4y - y^2 = x^2 + 2x + 2C_1$
Rearranging the terms to form the general solution:
$x^2 + y^2 + 2x - 4y + C = 0$,where $C = 2C_1$.
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