1. ## Taylor Series Solution

Consider the intial value problem:

$x^2y''(x) - y(x) = 1 - x$, y(1) = 1, y'(1) = -1

a) Identify the coefficients of the Taylor Polynomial of order of 4 around $x_0 = 1$ by successive differentiation of the DE. State the resulting polynomial approximation to the solution, $T_4(x) =$

I've only had a one hour lecture on Taylor series solutions so I'm pretty new at this and have not had a non homogeneous problem before and also have not had one that was not centered at zero. If someone could give me a hint or maybe an example that would be awesome.

2. Hi. The Taylor series would be $y(x)=\sum_{n=0}^{\infty} a_n(x-1)^n=\sum_{n=0}^{\infty} \frac{y^{(n)}(1)}{n!}(x-1)^n$

but $y(1)=1$ so $a_0=1$ and $y'(1)=-1$ so $a_1=-1$ and from the DE, $y''(1)=\frac{1-1+y(1)}{1}=-1$ so $a_2=-1/2$. Now, continue turning the crank.