Solve the given differential equation. \( e^{x^{2}} d y=x \sqrt{3-y} d x \) Choose the correct answer below. A. \( 3 \sqrt{4-y}=e^{-x^{2}}+C \) B. \( y=3-\frac{e^{-x^{4}}}{4}+C \) C. \( 4 \sqrt{3-y}=e^{-x^{2}}+C \) D. \( \sqrt{3-y}=4 e^{-x^{2}}+C \)

To solve the given differential equation, we can rearrange it as follows: \[ \frac{dy}{\sqrt{3 - y}} = x e^{-x^2} dx \] Now, integrating both sides: The left-hand side becomes: \[ \int \frac{dy}{\sqrt{3 - y}} = -2\sqrt{3 - y} \] The right-hand side requires integration by parts or using substitution (the integral of \(xe^{-x^2}\) is \(-\frac{1}{2} e^{-x^2}\)): \[ \int x e^{-x^2} dx = -\frac{1}{2} e^{-x^2} + C \] Putting it all together, we derive the relationship: \[ -2\sqrt{3 - y} = -\frac{1}{2} e^{-x^2} + C \] To put in simpler form, we can multiply through by -1 and rearrange: \[ 2\sqrt{3 - y} = \frac{1}{2} e^{-x^2} - C \] Looking for a match among the answer options, we have: \[ 4\sqrt{3 - y} = e^{-x^2} + C' \quad (\text{where } C' = -4C \text{ is a constant}) \] Thus, the correct answer is: C. \( 4 \sqrt{3-y}=e^{-x^{2}}+C \) --- Here's a bit more about differential equations in a fun way: These equations are like puzzles that can help us model real-world phenomena! Each piece of the puzzle (or equation) gives us a little clue about how one variable changes with respect to another. Think about how we can use them to understand everything from population growth to the motion of planets—fascinating, right? Additionally, while solving these equations, remember to watch out for algebraic slips! The most common mistake is dropping negative signs or forgetting to properly integrate both sides. Always double-check your integration to make sure you're not steering off course!