Question

\[\sqrt{a + x} dy + x\ dx = 0\]

Answer 1

We have, 
\[\sqrt{a + x}dy + x\ dx = 0\]
\[ \Rightarrow \sqrt{a + x}dy = - xdx\]
\[ \Rightarrow dy = \frac{- x}{\sqrt{a + x}}dx\]
\[ \Rightarrow dy = - \frac{\left( x + a - a \right)}{\sqrt{a + x}}dx\]
\[ \Rightarrow dy = - \left( \sqrt{a + x} - \frac{a}{\sqrt{a + x}} \right)dx\]
Integrating both sides, we get
\[\int dy = - \int\left( \sqrt{a + x} - \frac{a}{\sqrt{a + x}} \right)dx\]
\[ \Rightarrow y = - \frac{2 \left( a + x \right)^\frac{3}{2}}{3} + 2a\sqrt{a + x} + C\]
\[ \Rightarrow y + \frac{2}{3} \left( a + x \right)^\frac{3}{2} - 2a\sqrt{a + x} = C\]
\[\text{ Hence, }y + \frac{2}{3} \left( a + x \right)^\frac{3}{2} - 2a\sqrt{a + x} = \text{C is the solution to the given differential equation.}\]