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Question
Solve the following differential equation:
(xy2 + 2x) dx + (x2 y + 2y) dy = 0
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Solution
We have,
\[\left( x y^2 + 2x \right) dx + \left( x^2 y + 2y \right) dy = 0\]
\[ \Rightarrow x\left( y^2 + 2 \right) dx + y\left( x^2 + 2 \right) dy = 0\]
\[ \Rightarrow x\left( y^2 + 2 \right) dx = - y\left( x^2 + 2 \right) dy\]
\[ \Rightarrow \frac{x}{\left( x^2 + 2 \right)} dx = - \frac{y}{\left( y^2 + 2 \right)} dy\]
Integrating both sides, we get
\[\int\frac{x}{x^2 + 2} dx = - \int\frac{y}{y^2 + 2} dy\]
\[ \Rightarrow \frac{1}{2}\int\frac{2x}{x^2 + 2} dx = - \frac{1}{2}\int\frac{2y}{y^2 + 2} dy\]
\[ \Rightarrow \frac{1}{2}log \left| x^2 + 2 \right| = - \frac{1}{2}log \left| y^2 + 2 \right| + log C\]
\[ \Rightarrow \frac{1}{2}log \left| x^2 + 2 \right| + \frac{1}{2}log \left| y^2 + 2 \right| = log C\]
\[ \Rightarrow log \left| x^2 + 2 \right| + log \left| y^2 + 2 \right| = 2log C\]
\[ \Rightarrow log \left( \left| x^2 + 2 \right|\left| y^2 + 2 \right| \right) = log C^2 \]
\[ \Rightarrow \left( \left| x^2 + 2 \right|\left| y^2 + 2 \right| \right) = C^2 \]
\[ \Rightarrow \left( x^2 + 2 \right)\left( y^2 + 2 \right) = K\]
\[ \Rightarrow y^2 + 2 = \frac{K}{x^2 + 2}\]
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