The given differential equation is
\[
x y^{\prime} + y = \sqrt{x}
\]
We can rewrite it in standard form by dividing through by \( x \):
\[
y^{\prime} + \frac{1}{x} y = \frac{\sqrt{x}}{x} = x^{-1/2}
\]
From the rewritten equation, we identify:
\[
P(x) = \frac{1}{x}, \quad Q(x) = x^{-1/2}
\]
Using the integrating factor method, we find the integrating factor \( \mu(x) \):
\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = x
\]
Multiplying the entire equation by the integrating factor:
\[
x y^{\prime} + y = \sqrt{x} \implies \frac{d}{dx}(x y) = x^{1/2}
\]
Integrating both sides:
\[
\int \frac{d}{dx}(x y) \, dx = \int x^{1/2} \, dx
\]
This gives us:
\[
x y = \frac{2}{3} x^{3/2} + C
\]
Rearranging the equation, we find the general solution:
\[
y = \frac{2}{3} x^{1/2} + \frac{C}{x}
\]
We need to compare our general solution with the provided options:
A. \( 3xy = 2\sqrt{x} + C \)
B. \( 3y + C = 2x\sqrt{x} \)
C. \( 3\frac{x}{y} = C + 2x\sqrt{x} \)
D. \( 3xy = C + 2x\sqrt{x} \)
From our solution, we can rearrange it to match option A:
\[
3xy = 2\sqrt{x} + C
\]
The correct answer is
\(\boxed{A}\)
Answered
