Write out the first few terms of the Picard iteration method for the initial value problem $x'=x+2, x(0)=2$. Picard iteration says $u_0(t)=x_0, u_{k+1}(t)=x_0+\int_0^tf(u_k(s))ds$.
The first few terms I got are:
$u_0=x_0=2$
$u_1=x_0+\int_0^t (u_0(s)+2)ds = 2+[4s]_0^t=2+4t$
$u_2 = x_0+\int_0^t (u_1(s)+2)ds = 2+[4s+2s^2]_0^t=2+4t+2t^2$
I looked up the solution online and seem to get different results. What am I doing wrong? I have seen some solutions say that $u_0=x_0+2=4$, and while that does make the answer simple and look right, I don't know why this would be the case since any other example I have seen just takes $u_0=x_0$.
Best Answer
$$u_2 = 2+[4s+2s^2]_0^t=2+4t+2t^2$$ $$x(t)=4-2+4t+4\dfrac {t^2}2+.....$$ $$x(t)=-2+4\left (1+t+\dfrac {t^2}{2}+......\right)$$ $$x(t)=-2+4e^t$$ This looks like the correct answer.