Question 06 (a)
The height of fluid, in a cylindrical container is controlled by a pump so that it satisfies the differential equation
where , are constants, and is the initial height of the fluid.
(a) Solve this differential equation to determine the height at any later time .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Try separation of variables.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
Using separation of variables we can write
Both integrals are solved straight forward to yield
which we solve for h to obtain
where C2 is another constant.
All that's left to do now is to use the initial condition h(0) = h0 to solve for C2:
Therefore the height h(t) at any time t is given by
To double-check your answer you can take the derivative of h(t) and check that the given differential equation is indeed satisfied: