Science:Math Exam Resources/Courses/MATH103/April 2015/Question 09
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Question 09 

Suppose the power series solves the differential equation with the initial condition . Determine , , , and . 
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 

Substitute the given into the equation, and note that the index of summation can be shifted as the following rule:

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. Since , we can easily get from the initial condition. To get , , and , plug into the equation. Then, using Hint 1, we have
and
so that the equation can be written as
Now comparing each coefficient of on both side, we get
Using , the second equation implies that . On the other hand, plugging into the third equation, we obtain . Therefore, the answer is . 