Category: Basic Concepts

Introduction to Differential Equations

To put it simply, a differential equation is any equation that contains derivatives.

For example, $\frac{{{y}''}}{{t}}+{y}'+{t}{y}={0}$ and ${{\left({y}'''\right)}}^{{4}}+\sqrt{{{y}''}}-{y}'={5}{t}$ are both differential equations.

Existence and Uniqueness of the Solution to the ODE

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

Theorem 1. Consider the nth-order linear differential equation ${{y}}^{{{\left({n}\right)}}}+{p}_{{1}}{\left({t}\right)}{{y}}^{{{\left({n}-{1}\right)}}}+{p}_{{2}}{\left({t}\right)}{{y}}^{{{\left({n}-{2}\right)}}}+\ldots+{p}_{{n}}{\left({t}\right)}={f{{\left({t}\right)}}}$. If all coefficients ${p}_{{1}}{\left({t}\right)}$, ${p}_{{2}}{\left({t}\right)}$, ..., ${p}_{{n}}{\left({t}\right)}$ and ${f{{\left({t}\right)}}}$ are continuous on the interval ${\left({a},{b}\right)}$, the equation has the unique solution which satisfies the given initial conditions ${y}{\left({t}_{{0}}\right)}={y}_{{0}}$, ${y}'{\left({t}_{{0}}\right)}={{y}_{{0}}^{'}}$, ..., ${{y}}^{{{\left({n}-{1}\right)}}}{\left({t}_{{0}}\right)}={{y}_{{0}}^{{{\left({n}-{1}\right)}}}}$, where ${t}_{{0}}$ belongs to the interval ${\left({a},{b}\right)}$.