Find the Error: Solving Homework Problems

[Harmony]

Homework Statement

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Homework Equations

The Attempt at a Solution

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May I know which part of my solution went wrong? In the end the answer gives x/2 instead of 0.

 

[gabbagabbahey]

May I know which part of my solution went wrong? In the end the answer gives x/2 instead of 0.

Since $$y' (x)$$ contains only odd powers of x, the n=1 term does not vanish when you take the derivative of the series. Therefore,

$$y'' (x)= \sum_{n=1}^{\infty} \frac{(-1)^n (2n)(2n-1) x^{2n-2}}{4^n (n!)^2}$$

The extra $$x/2$$ in your answer will be canceled by the $$n=1$$ term.