Use a taylor series solution, centered at zero to solve the initial-value problem below. (find a 5th degree approximation only) dy/dx = x + y y(0) = 1

We're looking for a solution of the form

Given that , we would end up with .

Its first derivative is

The shifting of the index here is useful in the next step. Substituting these series into the ODE gives

Both series start with the same-degree term , so we can condense the left side into one series.

Pull out the first two terms ( and ) of the series:

Matching the coefficients of the and terms on either side tells us that

We know that , so and . The rest of the coefficients, for , are given according to the recurrence,

so that , , and . So the 5th degree approximation to the solution to this ODE centered at is

1/5^2 = 7.56/45