Chapter 8 - Mathematical Modeling With Differential Equations - 8.2 Separation Of Variables - Exercises Set 8.2 - Page 575: 5

$$ 2\ln {y}+ y^2 = e^x+c$$

Given $$(2+2y^2)y'= e^xy $$ Separate variables \begin{align*} 2\int \frac{1+y^2}{y}dy &= \int e^x dx \\ 2\int \left(\frac{1 }{y}+y\right)dy&=e^x+c\\ 2\left(\ln {y}+\frac{1}{2}y^2\right)&= e^x+c \end{align*}