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Variable Separable Method

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Question

Find the general solution of the differential equation
$(x + 2) \frac{d y}{d x} = x^{2} + 5 x - 3 (x \neq 2)$

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Solution

$\begin{matrix} T h e g i v e n d i f f e r e n t i a l e q u a t i o n m a b e w r i t t e n a s \frac{d y}{d x} = \frac{x^{2} + 5 x - 3}{x + 2} \Rightarrow d y = (\frac{x^{2} + 5 x - 3}{x + 2}) d x [s e p r a t i n g t h e v a r i b l e s] \Rightarrow \int d y = \int (\frac{x^{2} + 5 x - 3}{x + 2}) d x \Rightarrow y = \int {x + 3 - \frac{9}{(x + 2)}} d x + C [o n d i v i d i n g (x^{2} + 5 x - 3) b y (x + 2)] \Rightarrow y = \frac{x^{2}}{2} + 3 x - 9 log | x + 2 | + C . w h i c h i s t h e r e q u i r e d g e n e r a l s o l u t i o n . \end{matrix}$

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