Science:Math Exam Resources/Courses/MATH103/April 2017/Question 08 (a)
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Question 08 (a) 

Suppose the Taylor series solves the differential equation with the initial condition . (a) Calculate the first four coefficients 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 

Express the differential equation entirely in terms of Taylor series. Then collect like terms. 
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Solution 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since we have the power rule for differentiation, we can differentiate any Taylor series to obtain another Taylor series. With , we calculate
So
Because
becomes
Collecting like terms for the first three monomials on each side now gives
The first value can be determined from the initial condition because
It follows that . Using the above three equations, one can then calculate the following as the answer:

Solution 2 

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Please rate my easiness! It's quick and helps everyone guide their studies. We know that the coefficients of the Taylor series are given by and that the initial condition is Since solves the differential equation , or equivalently, , we must have By differentiating the equation for (with respect to ), we obtain , which implies that Differentiating the equation once more, we obtain , so We conclude that , , , and . 