Determine the first three nonzero terms of the taylor polynomial approximations for the given initial value problem. and y(0)=0

Note that with , the first sum reduces immediately (after taking out the first term in the sum) to , which implies that .

So we have




where we can expand as






Dropping all terms with order greater than 2 (because this is enough to generate 3 nonzero terms) reduces the equation to





so that the Taylor polynomial approximating the solution is



Just to demonstrate that the result is reasonable, I've attached a plot of an approximate solution with higher accuracy (blue) and the one we found (orange) over the interval .