Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)

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Rewrite the given differential equation to isolate \( \frac{dy}{dx} \). Multiply both sides by \( x \) to get \( \frac{dy}{dx} = x y e^{x^{2}} + 2 x \sqrt{y} e^{x^{2}} \).

Observe the right-hand side and consider a substitution to simplify the equation. Since the terms involve \( y \) and \( \sqrt{y} \), let \( u = \sqrt{y} \), so that \( y = u^{2} \).

Express \( \frac{dy}{dx} \) in terms of \( u \) and \( \frac{du}{dx} \). Using the chain rule, \( \frac{dy}{dx} = 2 u \frac{du}{dx} \). Substitute this and \( y = u^{2} \) into the equation.

After substitution, the equation becomes \( 2 u \frac{du}{dx} = x u^{2} e^{x^{2}} + 2 x u e^{x^{2}} \). Divide both sides by \( u \) (assuming \( u \neq 0 \)) to simplify to \( 2 \frac{du}{dx} = x u e^{x^{2}} + 2 x e^{x^{2}} \).

Rewrite the equation as \( \frac{du}{dx} - \frac{x}{2} e^{x^{2}} u = x e^{x^{2}} \). This is a linear first-order differential equation in \( u \). Identify the integrating factor \( \mu(x) = e^{-\int \frac{x}{2} e^{x^{2}} dx} \) and proceed to solve for \( u(x) \).

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