Science:Math Exam Resources/Courses/MATH215/December 2013/Question 04 (c)
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Question 04 (c) 

Let be the Laplace transform of the solution to the following initial value problem: where is the function defined in (a): (c) Calculate the inverse Laplace transform of Y(s) obtained in (b) to find the solution, y(t), to the IVP in (b). 
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 

You can start by trying to find the inverse Laplace transform of and . These inverse transforms can be derived by looking at and . 
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. We have and we need to find . The most important thing here is finding a function such that . The key identity to realize is that . With this knowledge, since , we have and (note the factor of 2!). Then
We use one more identity that which allows us to invert the first term (using a=1) on the righthand side. We find
