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

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1. Rewrite the given differential equation to isolate \( \frac{dy}{dx} \). Multiply both sides by \( x \) to get \( \frac{dy}{dx} = x y e^{x^{2}} + 2 x \sqrt{y} e^{x^{2}} \).
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 \( y = u^{2} \).
3. 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.
4. 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 \( 2 \frac{du}{dx} = x u e^{x^{2}} + 2 x e^{x^{2}} \).
5. 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 \( u(x) \).

Key Concepts

Separable Differential Equations

A separable differential equation can be written as a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This technique simplifies solving by integrating each side independently.

Substitution Method

Substitution involves introducing a new variable to simplify the differential equation, especially when terms like √y or y appear together. This method transforms the equation into a more manageable form, often linear or separable.

Integration Techniques

Solving the separated or substituted differential equation requires integration. Familiarity with integrating exponential functions and powers, as well as applying integration constants, is essential to find the general solution.