# List of Notes - Category: Basic Concepts

## Introduction

Simply saying differential equation is any equation that contains derivatives. For example, (y'')/t+y'+ty=0 , (y''')^4+sqrt(y'')-y'=5t are all differential equations.

There can be used different notation: either y^((n)) or (d^ny)/(dt^n) .

## Existence and Uniqueness

This note contains some theorems that refer to existence and uniqueness of solution of ODE.

Theorem 1. Consider the linear differential equation of n-th order: y^((n))+p_1(t)y^((n-1))+p_2(t)y^((n-2))+...+p_n(t)=f(t) . If all coefficients p_1(t), p_2(t), ..., p_n(t) and f(t) are continuous in interval (a,b) then equation has unique solution, which satisfies given initial conditions y(t_0)=y_0 , y'(t_0)=y_0^' , ..., y^((n-1))(t_0)=y_0^((n-1)) , where t_0 belongs to the interval (a,b) .