Power Series Questions

Question

I'm confused on how to solve these types of questions when it comes to Power Series.

The question states: For the following differential equations, give guaranteed convergence intervals for the expansion of the solutions in powers of x.

Question 1:

y''[x] - y[x] = Sin[x] 
y[0] = 3
y'[0] = -1

Question 2:

(1 - x) y''[x] - y[x] = Sin[x]
y[0] = 3
y'[0] = -1

I understand how convergence intervals work and can use the Ratio test or Power Series Convergence Test when I actually see the expansions of the functions, but I'm having difficulty when the question is formed like the examples above.

Please add what your thoughts are about the question. As it stands the question doesn't fit the rules described in the help center. — Sahiba Arora, Jul 30 '17 at 19:37

score 1 · Accepted Answer · answered Jul 30 '17 at 19:48

If $y(x)=\sum_{n=0}^\infty a_n x^n$, then $y'(x)=\sum_{n=1}^\infty na_n x^{n-1}$ and $y''(x)=\sum_{n=2}^\infty n(n-1)a_nx^{n-2}$. Thus,

$$y''(x)-y'(x)=\sum_{n=0}^\infty (n+1)a_{n+1}x^n+\sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n=\sum_{n=0}^\infty \left[(n+1)(a_{n+1}+(n+2)a_{n+2})\right]x^n$$

Now equate the quantity between square brackets to each of the coefficients of the power series expansion of $\sin(x)$.

Power Series Questions

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