Differential equation:

1. given y'=y+x; y(0)=1.

(a) using picard's method, give the first five terms of the iterative solution starting with latex: y_0\left(x\right)=1 y 0 ( x ) = 1 .

(b) using picard's method, give the first five terms of the iterative solution starting with latex: y_0\left(x\right)=x y 0 ( x ) = x .

(c) based on your answers, do different initial guess functions lead to different outcomes?

(d) solve the equation exactly and use power series expansion to compare it with the iterative solutions in (a) and (b).