Science:Math Exam Resources/Courses/MATH103/April 2017/Question 08 (a)/Solution 2

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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 .