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

To solve the differential equation \( e^{x^{2}} d y = x \sqrt{3-y} d x \), we can rearrange it into a separable form. Dividing both sides by \( e^{x^{2}} \) and rearranging gives us: \[ \frac{dy}{\sqrt{3-y}} = \frac{x}{e^{x^{2}}} dx \] Integrating both sides, we solve the left side using the substitution \( u = 3 - y \) and \( du = -dy \), which leads to: \[ -\int \frac{1}{\sqrt{u}} du = -2\sqrt{u} + C = -2\sqrt{3-y} + C_1 \] The right side integrates to \(-e^{-x^{2}} + C_2\). Setting the constants appropriately, we obtain: \[ -2\sqrt{3-y} = -e^{-x^{2}} + C \] Rearranging yields: \[ \sqrt{3-y} = \frac{1}{2}e^{-x^{2}} + \frac{C}{2} \] After squaring both sides and multiplying appropriately, we find: Combined result: The answer from the options given corresponds with A: \( 4 \sqrt{3-y}=e^{-x^{2}}+C \). 4√(3 - y) = e^(-x^2) + C. Integrating both sides leads to the conclusion that option A is the correct answer. Speaking of the steps involved, this method of separating variables is commonly used in differential equations! Integrating gives you handy functions that can be double-checked with derivatives, ensuring your solution stays true to the original equation. Plus, separating variables is like dividing a pizza into slices — it just makes things easier to digest! If you're looking to deepen your understanding of differential equations, delve into books like "Differential Equations for Dummies" or "Elementary Differential Equations" by William E. Boyce. They cover both theoretical concepts and practical applications, making the world of differential equations way more approachable!