Find the general solution to the differential equation \frac{dy}{dt}=t^{3}+2t^{2}-8t A

David Troyer

David Troyer

Answered question


Find the general solution to the differential equation
Also, part 8B. asks: Show that the constant function y(t)=0 is a solution.

Answer & Explanation

Maria Lopez

Maria Lopez

Beginner2022-01-18Added 32 answers

Parts of problems that say ''show that [explicitly given function] is a solution'' can be solved by simply ''plugging in''. E.g., if you're supposed to show that y(t)=1tt is a solution to the differential equation dydt=y2, then you should compute dydt=1(1t)2, compute y2=1(1t)2, and ''plug them in'' to check that the differential equation is satisfied for this choice of y.
You're correct that the difference in the second problem is that you no longer have the derivative with respect to t expressed as a function of t. When the equation involves both a function y and its derivative (or derivatives), it can generally be more difficult to solve. In this case, a hint I'll give you is that dydt=1dtdy
You can first turn things around and think of t as a function of y. You can find t(y) using similar methods to the first problem (but with a more complicated function), and then find the inverse function to get y(t).
Janet Young

Janet Young

Beginner2022-01-19Added 32 answers

The difference is that the main variable is y and you separate the variables to integrate and obtain the solution
Hint 1:
The first equation is very easy, so:
Hint 2:
For the other.
Then proceed integrating.


Skilled2022-01-24Added 474 answers

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 dydx=x (which would mean that y=12x2+C), and a completely different thing if you are told dydx=y (this tells you that the function y is equal to its derivative; which means that y=Aex for some constant A). Actually, we can solve the second differential equation; but we dont

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