a) Give the first few terms in the power series expansion (up to the fourth power) of the solution of the initial value problem:
y' = e^x + x cos y , y(0) =0.
b) Determine first terms in the power series expansion y =[summation ]a(k)x^k for the solution to the initial value problem :
y' = x^3 + y^3 , y(0) = 1.
Determine the power series expansion of the solution u = [summation] b(k) x^k of
u' = u^3 , u(0) = 1 and show that a(k) >= b(k).
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