The given differential equation is:
\[\sqrt{a + x} \frac{dy}{dx} + xy = 0\]To solve this differential equation, we will use the integrating factor method. First, we rewrite the equation in the standard linear differential equation form:
\[\frac{dy}{dx} + \frac{x}{\sqrt{a + x}} y = 0\]The standard form of a first-order linear differential equation is:
\[\frac{dy}{dx} + P(x)y = Q(x)\]Here, \(P(x) = \frac{x}{\sqrt{a + x}}\) and \(Q(x) = 0\).
The integrating factor, \(\mu(x)\), is given by:
\[\mu(x) = e^{\int P(x) \, dx}\]Substitute \(P(x)\) into the integrating factor formula:
\[\mu(x) = e^{\int \frac{x}{\sqrt{a + x}} \, dx}\]Now, let's solve the integral:
\[\int \frac{x}{\sqrt{a + x}} \, dx\]Let \(u = \sqrt{a + x}\). Then, \(x = u^2 - a\) and \(dx = 2u \, du\).
Substitute these into the integral:
\[\int \frac{u^2 - a}{u} \cdot 2u \, du = 2 \int (u^2 - a) \, du = 2 \left(\frac{u^3}{3} - au\right)+ C\]We get:
\[\int \frac{x}{\sqrt{a + x}} \, dx = \frac{2}{3}(a + x)^{3/2} - 2a (a + x)^{1/2}+ C\]Return to the integrating factor:
\[\mu(x) = e^{\frac{2}{3}(a + x)^{3/2} - 2a (a + x)^{1/2}}\]Using the integrating factor, the solution to the differential equation is given by:
\[y \cdot \mu(x) = C\]So, the solution is:
\(y = Ce^{\frac{2}{3}(2a-x)\sqrt{x+a}}\)
This matches the given correct answer. Thus, the solution to the differential equation is:
\(y = Ce^{\frac{2}{3}(2a-x)\sqrt{x+a}}\)