Fundamentals of differential equations by Nagle Saff Snider, 8th edition, No. 26, Page 445

Question

find at least the first four nonzero terms in a power series expansion about for the solution to the given initial value problem.

$$ (x^2-x+1)y''-y'-y=0;\\~\\ y(0)=0, ~~ y'(0)=1 $$

I confused how to manipulate..

$$\sum_2^\infty n(n-1)a_nx^n$$

Answer 1

I assume your differential equation is $(x^2-x+1)y'' - y'- y = 0$ with the initial conditions $y(0)=0, y'(0) = 1$

The power series (Taylor/Maclaurin) of $y(x) = \frac{y(0)}{0!}x^0+\frac{y'(0)}{1!}x^1+\frac{y''(0)}{2!}x^2+...+\frac{y^{(r)}(0)}{r!}x^r+...$

They asked for the first four non-zero terms. Since $y(0) = 0$, that doesn't count. If every derivative up to and including $y^{(4)}$ is non-zero, you're done.

You can easily compute the second derivative directly from the equation by setting $x=0$ and using the given conditions.

You can then differentiate the original equation, and then differentiate again using product rule carefully, and set $x=0$ each time to work out the value of third and fourth derivatives at $x=0$. That should be sufficient. Then apply the Taylor expansion.