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)` .