Write out the first few terms of the picard iteration scheme for each of the following initial value problems. where possible, find explicit solutions and describe the domain of this solution.

a) x'=x+2 ; x(0)=2
b) x'=x^(4/3) ; x(0)=0
c) x'=x^(4/3) ; x(0)=1

Given an ODE with initial condition , the general process is to write the ODE as an integral equation,



By setting for all , we get the following recurrence for .



From this we work towards finding a pattern for so that we can find a solution of the form .



Write this as the integral equation,



First step:





Second step:





Third step:





Fourth step:





You should already start seeing a pattern. Recall that



Multiplying this by 2 gives



which matches the solution we have for except for that first term. So subtracting that, we find a solution of



with a domain of .

Hopefully this gives some insight on how to approach the other two problems.