First slide

Methods of solving first order first degree differential equations

Question

$Solution of the differential equation \frac{d y}{d x} = \frac{2 x - y + 1}{x + y + 2} is$

Moderate

A

$x y - y^{2} = 2 y + C$
B

$x y + y^{2} + 2 y = 3 x^{2} + C$
C

$x y + y^{2} = 2 y + 3 x^{2} + C$
D

$x y + \frac{y^{2}}{2} + 2 y = x^{2} + x + C$

Solution

$\begin{array}{l} Given that \frac{d y}{d x} = \frac{2 x - y + 1}{x + y + 2} \\ \Rightarrow (x + y + 2) d y = (2 x - y + 1) d x \end{array}$
$\begin{array}{l} \Rightarrow x d y + (y + 2) d y = 2 x d x - y d x + d x \\ \Rightarrow x d y + y d x + (y + 2) d y = 2 x d x + d x \\ \Rightarrow d (x y) + (y + 2) d y = 2 x d x + d x \end{array}$
Integrating on both sides
$\begin{array}{l} \Rightarrow x y + \frac{y^{2}}{2} + 2 y = x^{2} + x + C \\ Therefore, the correct answer is (4). \end{array}$

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