Help solving a differential equation my Calculus II class is nearing the end of the quarter and we've just started differential equations to get ready for Calculus III. In my homework, I came upon these problems.
One of the problems was:
Find the general solution to the differential equation 
$$\frac{dy}{dt} = t^3 + 2t^2 - 8t.$$
The teacher just said to integrate. So I did. Then in question 8a it gives the differential equation:
$$\frac{dy}{dt} = y^3 + 2y^2 - 8y.$$
and asks "Why can't we find a solution like we did to the previous problem? My guess was: "In 7 we were integrating with respect to t. Since this equation is the highest order derivative, we can't solve it like # 7". Although, I have no confidence in that answer and I'm not sure it makes total sense even to me.
Also, part 8B. asks: Show that the constant function $y(t) = 0$ is a solution.
I've done a problem like this before, except that it wasn't a constant function. This problem seems like a question that asks: "show that every member of the family of functions $y = (\ln x + C)/x$ is a solution to the differential equation (some diff. equation)" except it seems a little bit different.
Any hints on how I can solve this?
Thank you.

In the first question, you are given the derivative in terms of the variable. But in the second question, you are given an expression for the derivative that involves the function. For instance, it would be one thing if you were told $\frac{dy}{dx} = x$ (which would mean that $y = \frac{1}{2}x^2 + C$), and a completely different thing if you are told $\frac{dy}{dx}=y$ (this tells you that the function $y$ is equal to its derivative; which means that $y=Ae^x$ for some constant $A$). 
Actually, we can solve the second differential equation; but we don't solve it by simple integration, just like we don't solve the equation $y' = y$ by integrating. 
Showing that $y(t)=0$ is a solution to the second equation is just a question of plugging in $0$ for $y$ and verifying that you get a solution by verifying that the resulting equality is true.
You never said what question 8a was, through; were you asked to solve the differential equation, or just to say what is the difference between it and the previous one?
