Differential equation multivariable solution I don't understand how I would solve the following problem:

 Where does the $F(t,y) = -5$ come from? I tried solving it normally, do I create a multivariable function that satisfies $F(t,y) = -5$? Guidance would be appreciated, thanks. 

This DE is separable. Rewrite as
$$(9y^2-30e^{6y})\,dy=\frac{25t^4}{1+t^{10}}\,dt$$
and integrate.
On the left, we get $3y^3-5e^{6y}$. On the right, make the substitution $t^5=u$.
After a short while, we arrive at $5\arctan(t^5)+C$.  
Use the initial condition to find $C$. Since $y=0$ when $t=0$, we get $C=-5$. At this point, we more or less have to stop, for we cannot solve explicitly for $y$ in terms of $t$. (We can solve explicitly for $t$ in terms of $y$.)
So we end up with the equation $F(y,t)=-5$, where 
$$F(y,t)=3y^3-5e^{6y}-5\arctan(t^5).$$
