Question

\[\left( x + 2 \right)\frac{dy}{dx} = x^2 + 3x + 7\]

Answer 1

We have, 
\[\left( x + 2 \right)\frac{dy}{dx} = x^2 + 3x + 7\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x^2 + 3x + 7}{x + 2}\]
\[ \Rightarrow dy = \left( \frac{x^2 + 3x + 7}{x + 2} \right)dx\]
Integrating both sides, we get
\[\int dy = \int\left( \frac{x^2 + 3x + 7}{x + 2} \right)dx\]
\[ \Rightarrow \int dy = \int\left( \frac{x^2 + 3x + 2 + 5}{x + 2} \right)dx\]
\[ \Rightarrow \int dy = \int\left[ \frac{\left( x + 2 \right)\left( x + 1 \right) + 5}{x + 2} \right]dx\]
\[ \Rightarrow \int dy = \int\left( x + 1 + \frac{5}{x + 2} \right)dx\]
\[ \Rightarrow y = \frac{x^2}{2} + x + 5 \log\left| x + 2 \right| + C\]
\[\text{ So, } y = \frac{x^2}{2} + x + 5 \log\left| x + 2 \right| +\text{C is defined for all } x \in R\text{ except }x = - 2 . \]
\[\text{Hence, }y = \frac{x^2}{2} + x + 5 \log\left| x + 2 \right| + \text{C, where }x \in R - \left\{ 2 \right\},\text{ is the solution to the given differential equation.}\]