Chapter 8 - Mathematical Modeling With Differential Equations - 8.2 Separation Of Variables - Exercises Set 8.2 - Page 575: 4

$$y =\sqrt{ \frac{2}{3}\ln (1+x^4)+2c}$$

Given $$(1+x^4)\frac{dy}{dx}= \frac{x^3}{y} $$ Separate variables \begin{align*} \int ydy &= \int \frac{x^3}{1+x^4} dx \\ \frac{1}{2}y^2&=\frac{1}{3}\ln (1+x^4)+c\\ y^2&= \frac{2}{3}\ln (1+x^4)+2c\\ y&=\sqrt{ \frac{2}{3}\ln (1+x^4)+2c} \end{align*}