# Ordinary Differential Equations/Successive Approximations

${\displaystyle y'=f(x,y)}$ has a solution ${\displaystyle y}$ satisfying the initial condition ${\displaystyle y(x_{0})=y_{0}}$, then it must satisfy the following integral equation:

${\displaystyle y=y_{0}+\int _{x_{0}}^{x}f(t,y(t))dt}$

Now we will solve this equation by the method of successive approximations.

Define ${\displaystyle y_{1}}$ as:

${\displaystyle y_{1}=y_{0}+\int _{x_{0}}^{x}f(t,y_{0})dt}$

And define ${\displaystyle y_{n}}$ as

${\displaystyle y_{n}=y_{0}+\int _{x_{0}}^{x}f(t,y_{n-1})dt}$

We will now prove that:

1. If ${\displaystyle f(x,y)}$ is bounded and the Lipschitz condition is satisfied, then the sequence of functions converges to a continuous function
2. This function satisfies the differential equation
3. This is the unique solution to this differential equation with the given initial condition.

## Proof

First, we prove that ${\displaystyle y_{n}}$ lies in the box, meaning that ${\displaystyle |y_{n}(x)-y_{0}|<{\frac {1}{2}}h}$. We prove this by induction. First, it is obvious that ${\displaystyle |y_{1}(x)-y_{0}|\leq {\frac {1}{2}}h}$. Now suppose that ${\displaystyle |y_{n-1}(x)-y_{0}|\leq {\frac {1}{2}}h}$. Then ${\displaystyle |f(t,y_{n-1}(t))|\leq M}$ so that

${\displaystyle |y_{n}(x)-y_{0}|\leq \int _{x_{0}}^{x}|f(t,y_{n-1}(t))|dt\leq M(x-x_{0})\leq {\frac {1}{2}}Mw\leq {\frac {1}{2}}h}$. This proves the case when ${\displaystyle x_{0}, and the case when ${\displaystyle x is proven similarily.

We will now prove by induction that ${\displaystyle |y_{n}(x)-y_{n-1}(x)|<{\frac {MK^{n-1}}{n!}}(x-x_{0})^{n}}$. First, it is obvious that ${\displaystyle |y_{1}(x)-y_{0}|. Now suppose that it is true up to n-1. Then

${\displaystyle |y_{n}(x)-y_{n-1}(x)|\leq \int _{x_{0}}^{x}|f(t,y_{n-1}(t))-f(t,y_{n-2}(t))|dt<\int _{x_{0}}^{x}K|y_{n-1}(t)-y_{n-2}(t)|dt}$ due to the Lipschitz condition.

Now,

${\displaystyle |y_{n}(x)-y_{n-1}(x)|<{\frac {MK^{n-1}}{(n-1)!}}\int _{x_{0}}^{x}||u-x_{0}|^{n-1}du={\frac {MK^{n-1}}{n!}}|x-x_{0}|^{n}}$.

Therefore, the series of series ${\displaystyle y_{0}+\sum _{n=1}^{\infty }(y_{n}(x)-y_{n-1}(x))}$ is absolutely and uniformly convergent for ${\displaystyle |x-x_{0}|\leq {\frac {1}{2}}w}$ because it is less than the exponential function.

Therefore, the limit function ${\displaystyle y(x)=y_{0}+\sum _{n=1}^{\infty }(y_{n}(x)-y_{n-1}(x))=\lim _{n\rightarrow \infty }y_{n}(x)}$ exists and is a continuous function for ${\displaystyle |x-x_{0}|\leq {\frac {1}{2}}w}$.

Now we will prove that this limit function satisfies the differential equation.