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.