Chapter 8 - Mathematical Modeling With Differential Equations - 8.4 First-Order Differential Equations And Applications - Exercises Set 8.4 - Page 592: 9

$$y= -1+4e^{x^2 }$$

Given $$ \frac{dy}{dx} - 2xy =2x,\ \ \ \ \ y(0)=3$$ Since \begin{align*} \mu(x)&=e^{\int -2x dx}\\ &=e^{-x^2 } \end{align*} Then \begin{align*} y\mu(x)&=\int \mu(x) q(x)dx\\ e^{-x^2 } y&=\int 2xe^{-x^2 } dx\\ &= -e^{-x^2 } +c \end{align*} Since $y(0)=3,$ then $c=4$ Hence $$y= -1+4e^{x^2 }$$