The method

Suppose we have an ODE (with initial conditions)

\[\begin{equation} x'\!\left(t\right)=f\!\left(x\!\left(t\right),t\right), \quad x\!\left(t_0\right)=C, \label{eq: ode} \end{equation}\]

where $x$ is the unknown function, and $f$ is Lipschitz continuous in its first argument and continuous in its second argument. By Picard–Lindelöf theorem, we can seek the unique solution within $t\in\left[t_0-\varepsilon,t_0+\varepsilon\right]$.

Here, I propose the following method: we can write out a sequence of functions defined by

\[\begin{equation}x_0\!\left(t\right):=C,\label{eq: x0}\end{equation}\] \[\begin{equation} x_{n+1}\!\left(t\right):=\int_{t_0}^tf\!\left(x_n\!\left(s\right),s\right)\,\mathrm ds+C \label{eq: xn+1} \end{equation}\]

(the properties of $f$ guarantee that the integral is well-defined). Then, if the sequence $\left(x_n’\right)$ converges uniformly on $\left[t-\varepsilon,t+\varepsilon\right]$ (question: can this condition actually be proved?), then the sequence $\left(x_n\right)$ converges uniformly to a function $x$ on $\left[t-\varepsilon,t+\varepsilon\right]$, which is the unique solution to Equation \ref{eq: ode}.

Proof

The proof is easy. Note from Equation \ref{eq: xn+1} that

\[x_{n+1}'\!\left(t\right)=f\!\left(x_n\!\left(t\right),t\right).\]

Then, take the limit $n\to\infty$. By the uniform convergence, we recovers Equation \ref{eq: ode}.

An example

Suppose $f\!\left(x,t\right):=x$ and $t_0:=0$, then

\[x_n\!\left(t\right)=C\sum_{j=0}^{n-1}\frac{t^j}{j!}.\]

This is because

\[\int_0^tx_n\!\left(s\right)\,\mathrm ds+C=C\sum_{j=0}^{n-1}\frac{t^{j+1}}{j!\left(j+1\right)}+C=x_{n+1}\!\left(t\right).\]

By taking the limit $n\to\infty$, we get $x\!\left(t\right)=C\mathrm e^t$, which means that the unique solution to $x’\!\left(t\right)=x\!\left(t\right)$ with initial condition $x\!\left(0\right)=C$ is $x\!\left(t\right)=C\mathrm e^t$, expectedly.

Approximation

This method is good because integration is sometimes much easier than solving ODE. We can use integration to get functions that are close to the exact solution. A natural question to ask is how close $x_n$ is to the exact solution $x$.

If Equation \ref{eq: ode} is defined on $\mathbb R^n$ (where we may define differences and infinitesimals), it is clear that $x_n-x$ is an infinitesimal of higher order than $\left(t-t_0\right)^n$.