Find an equation of the curve that satisfies dy/dx=8x^7y and whose y-intercept is 6.
The function evaluated at $\displaystyle x=0$ implies that $\displaystyle y=6$. Thus, $\displaystyle y(0)=6$.
$\displaystyle y' = 8x^7 y$ note $\displaystyle y\not = 0$,
$\displaystyle \frac{y'}{y} = 8x^7$
$\displaystyle \int \frac{y'}{y} dx = \int 8x^7dx$
$\displaystyle \ln |y| = x^8 + C$
$\displaystyle y = Ce^{x^8} \mbox{ where }C>0$
In order to satisfy the IVP we need $\displaystyle C=6$.