# Category: Basic Concepts

## Introduction to Differential Equations

To put it simply, a differential equation is any equation that contains derivatives. For example, (y'')/t+y'+ty=0 and (y''')^4+sqrt(y'')-y'=5t are both differential equations.

Different notations can be used: either y^((n)) or (d^ny)/(dt^n).

## Existence and Uniqueness of the Solution the ODE

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

Theorem 1. Consider the n-th-order linear differential equation: 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 on the interval (a,b), the equation has the unique solution which satisfies the 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).