Chapter 6.5, Problem 24P

The Method of Successive Approximations. Consdier the initial value problem

$\text{x'=Ax,}\text{x}\left(0\right){\text{=x}}_{\text{0}}$,         (i)

where $\text{A}$ is a constant matrix and ${\text{x}}_{\text{0}}$ is prescribed vector.

(a) Assuming that a solution $\text{x=}\varphi \left(t\right)$ exists, show that it must satisfy the integra; equation

$\varphi \left(t\right)={\text{x}}_0+{\displaystyle \underset{0}{\overset{t}{\int }}\text{A}}\varphi \left(s\right)ds$.         (ii)

(b) Start with the initial approximation ${\varphi }^{\left(0\right)}={\text{x}}_0$. Substitute the expression for $\varphi \left(t\right)$ in the right hand side of Eq.(ii) and obtain a new approximation ${\varphi }^{\left(1\right)}\left(t\right)$. Show that ${\varphi }^{\left(1\right)}\left(t\right)\text{=}\left({\text{I}}_{\text{n}}\text{+A}t\right){\text{x}}_{\text{0}}$.         (iii)

(c) Repeat the process and thereby obtain a sequence of approximations ${\varphi }^{\left(0\right)},{\varphi }^{\left(1\right)},{\varphi }^{\left(2\right)},\mathrm{......},{\varphi }^{\left(n\right)},\mathrm{..}$ Use an inductive argument to show that

${\varphi }^{\left(n\right)}\left(t\right)=\left({\text{I}}_{\text{n}}\text{+A}t+{\text{A}}^2\frac{t^2}{2!}+\mathrm{......}+{\text{A}}^n\frac{t^n}{n!}\right){\text{x}}_{\text{0}}$.         (iv)

(d) Let $n\to \infty$ and show that the solution of the initial value problem (i) is

$\varphi \left(t\right)=e^{\text{At}}{\text{x}}_{\text{0}}$         (v)