Question

Solve the differential equation in Exercises 9–22.21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)

Accepted Answer

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 $$ u 
eq 0 ) to $$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 $$ u(x) .$$

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

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Key Concepts

Separable Differential Equations

Substitution Method

Integration Techniques