Find the general solution to the differential equation.
$\frac{dy}{dx}=y\sqrt{x}$

• A. $y=\frac23x^{\frac32}+C$
• B. $y=\frac32x^{\frac32}+C$
• C. $y=Ce^{\frac23x^{\frac32}}$
• D. $y=Ce^{\frac23x^{\frac23}}$

Answer: (C)

Verified step by step guidance

1. Step 1: Start by identifying the type of differential equation. The given equation is a first-order differential equation of the form \( \frac{dy}{dx} = y \sqrt{x} \). This suggests that it may be separable, meaning we can rewrite it to separate the variables \( y \) and \( x \).
2. Step 2: Rewrite the equation to separate the variables. Divide both sides by \( y \) and multiply by \( dx \) to get \( \frac{1}{y} dy = \sqrt{x} dx \). This allows us to integrate each side independently.
3. Step 3: Integrate both sides. The left-hand side becomes \( \int \frac{1}{y} dy = \ln|y| \), and the right-hand side becomes \( \int \sqrt{x} dx = \frac{2}{3}x^{\frac{3}{2}} + C_1 \), where \( C_1 \) is the constant of integration.
4. Step 4: Combine the results of the integration. We now have \( \ln|y| = \frac{2}{3}x^{\frac{3}{2}} + C_1 \). Exponentiate both sides to solve for \( y \), yielding \( y = e^{\frac{2}{3}x^{\frac{3}{2}} + C_1} \). Since \( e^{C_1} \) is just a constant, we can write \( y = Ce^{\frac{2}{3}x^{\frac{3}{2}}} \), where \( C \) is a constant.
5. Step 5: Verify the solution. Substitute \( y = Ce^{\frac{2}{3}x^{\frac{3}{2}}} \) back into the original differential equation \( \frac{dy}{dx} = y \sqrt{x} \) to confirm that it satisfies the equation. This ensures the solution is correct.