Science:Math Exam Resources/Courses/MATH103/April 2010/Question 06 (a)
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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! 
Hint 

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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using separation of variables we can write as Both integrals are solved straight forward to yield which we solve for h to obtain where C_{2} is another constant. All that's left to do now is to use the initial condition h(0) = h_{0} to solve for C_{2}: Therefore the height h(t) at any time t is given by
To doublecheck your answer you can take the derivative of h(t) and check that the given differential equation is indeed satisfied: 