Picard iteration.

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

$$x'=x+2;\:x(0)=2$$

Question

Picard iteration. 

Write out the first few terms of the Picard iteration scheme for the following initial value problems. If possible, find explicit solutions and describe the domain of this solution.

$$x'=x+2;\:x(0)=2$$

richyw · Answer

so I can just plug it into the formula$$u_{k+1}(t)=x_0+\int^t_0f(u_k(s))ds$$

richyw · Answer

but I don't really understand what I am doing and then I have no idea how I get an explicit solution from this.

richyw · Answer

like I get$$u_0=2$$$$u_1=2+4t$$$$u_2=2t^2+4t+2$$$$u_3=\frac{2}{3}t^3+2t^2+4t+2$$

anonymous · Answer

What the steps tend to spit out is the expansion of the solution. You keep going until you recognize the expansion as one of a function you know or you just write it as an infinite series.

goformit100 · Answer

the fifth step

richyw · Answer

does anyone know of a video that explains this briefly?

richyw · Answer

my book doesn't it was written by a smart guy but a terrible teacher. has cardboard paper to make up for the lack of content haha!

anonymous · Answer

www.math.msu.edu/~seal/teaching/f09/picard_iteration.pdf
There is even a good example there

richyw · Answer

thanks!