The method of successive approximations (see Section \(2.8)\) can also be
applied to systems
of equations. For example, consider the initial value problem
$$
\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}, \quad \mathbf{x}(0)=\mathbf{x}^{0}
$$
where \(\mathbf{A}\) is a constant matrix and \(\mathbf{x}^{0}\) a prescribed
vector.
(a) Assuming that a solution \(\mathbf{x}=\Phi(t)\) exists, show that it must
satisfy the integral equation
$$
\Phi(t)=\mathbf{x}^{0}+\int_{0}^{t} \mathbf{A} \phi(s) d s
$$
(b) Start with the initial approximation \(\Phi^{(0)}(t)=\mathbf{x}^{0} .\)
Substitute this expression for \(\Phi(s)\) in the right side of Eq. (ii) and
obtain a new approximation \(\Phi^{(1)}(t) .\) Show that
$$
\phi^{(1)}(t)=(1+\mathbf{A} t) \mathbf{x}^{0}
$$
(c) Reppeat this process and thereby obtain a sequence of approximations
\(\phi^{(0)}, \phi^{(1)}\),
\(\phi^{(2)}, \ldots, \phi^{(n)}, \ldots\) Use an inductive argument to show
that
$$
\phi^{(n)}(t)=\left(1+A t+A^{2} \frac{2}{2 !}+\cdots+A^{x} \frac{r^{2}}{n
!}\right) x^{0}
$$
(d) Let \(n \rightarrow \infty\) and show that the solution of the initial value
problem (i) is
$$
\phi(t)=\exp (\mathbf{A} t) \mathbf{x}^{0}
$$
