Science:Math Exam Resources/Courses/MATH103/April 2014/Question 07 (c)
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Question 07 (c) 

Suppose the function given by a power series solves following differential equation
with the initial value . Determine and . Hint: Plug the series into the differential equation. Work must be shown for full marks. Simplify fully. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

After plugging the series into the differential equation, differentiate the lefthand side termbyterm. 
Hint 2 

Following the first hint yields an equation involving the equality of two series. Compare the coefficients of the monomials in the two series. 
Hint 3 

Don’t forget to use the given initial condition. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. Substituting into yields
Writing out the first few terms explicitly on both sides, we have
Comparing the coefficients of on both sides, we see that . But comparing the coefficients of on both sides, we see that . Thus, , i. e. . Now comparing the coefficients of on both sides, we get . Since does not appear on the lefthand side, we determine it instead by using the given initial condition . That is, setting in , we get . Thus, implies . Putting this all together, we get , i. e. . 