
Taylor Series Solution
Consider the intial value problem:
$\displaystyle x^2y''(x)  y(x) = 1  x$, y(1) = 1, y'(1) = 1
a) Identify the coefficients of the Taylor Polynomial of order of 4 around $\displaystyle x_0 = 1$ by successive differentiation of the DE. State the resulting polynomial approximation to the solution, $\displaystyle 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.

Hi. The Taylor series would be $\displaystyle y(x)=\sum_{n=0}^{\infty} a_n(x1)^n=\sum_{n=0}^{\infty} \frac{y^{(n)}(1)}{n!}(x1)^n$
but $\displaystyle y(1)=1$ so $\displaystyle a_0=1$ and $\displaystyle y'(1)=1$ so $\displaystyle a_1=1$ and from the DE, $\displaystyle y''(1)=\frac{11+y(1)}{1}=1$ so $\displaystyle a_2=1/2$. Now, continue turning the crank. :)