Basic definition and local existence of a unique solution

 

Notation

WLOG : without loss of generality

LHS : Left hand side

RHS : Right hand side

IVP : Initial value problem

BVP : Boundary value problem

 

 

 

$a_{n}(x)\frac{\mathrm{d}^{n}y}{\mathrm{d} x^{n}} + a_{n-1}(x)\frac{\mathrm{d}^{n-1}y}{\mathrm{d} x^{n-1}} + \cdots + a_{1}(x)\frac{\mathrm{d}y}{\mathrm{d} x} + a_{0}(x)y = g(x)$          $(*)$

 

Def 1

Any function $\phi$, defined on an interval $I$ and possessing at least $n$ derivatives that are continuous on $I$, is said to be a solution of a differential equation $(*)$ on the interval $I$ if $\phi(x)$ reduces to an identity when substituted into a differential equation $(*)$.

 

 

 

Def 2

A relation $G(x, y) = 0$ is said to be an implicit solution of a differential equation $(*)$ on an interval $I$, provided that there exists at least one function $\phi$ that satisfies the relation as well as the differential equation $(*)$ on $I$.

 

 

 

Thm 1 local existence of a unique solution

Let $R$ be a rectangular region in the $xy$-plane($\mathbb{R}^{2}$) defined by $a \leq x \leq b, \; c \leq y \leq d$ that contains the point $(x_{0}, y_{0})$ in the interior. i.e.

$(x_{0}, y_{0}) \in R = \left\{(x, y) \in \mathbb{R}^{2} \; | \; a \leq x \leq b, \; c \leq x \leq d \right\}$

If $f(x, y)$ and $\frac{\partial f}{\partial y}$ are continous on $R$, then there exists some interval $I_{0} = (x_{0} - h, x_{0} + h ), \; h>0$, contained in $[a, b]$, and a unique function $y(x)$ defined on $I_{0}$  s.t.  $y(x)$ is a solution of given IVP

$\left\{\begin{matrix}
\frac{\mathrm{d}y}{\mathrm{d} x} = f(x, y) \\
y(x_{0}) = y_{0} \end{matrix}\right. $