Differential equations are mathematical expressions involving an unknown function and its derivatives, essential for describing many physical phenomena. Ordinary differential equations (ODEs) focus on functions of a single independent variable and their derivatives. A notable type of ODE is the sep...
Definition and Types of Differential Equations
kosala Madhushanka
September 2024
1 Introduction
Differential equations are equations that involve an unknown function and its
derivatives. They are used to formulate, model, and solve problems that involve
functions and their rates of change.
There are two main types of differential equations:
1.ordinary differential equations (ODEs).
2.partial differential equations (PDEs).
ODEs involve functions of a single variable and their derivatives. PDEs
involve functions of multiple variables and their partial derivatives.
This document focuses on ODEs and their classification based on order and
linearity.
1.1 Order of a Differential Equation
The order of a differential equation is the highest order of the derivative that
appears in the equation.
Examples:
dy
* dx + y = x is a first-order ODE.
2
d y
* dx 2 + sin(y) = 0 is a second-order ODE.
Types of Differential Equations Based on linearity, ODEs can be classified
into:
Linear Differential Equations:
dn y dn−1 y
An ODE is linear if it can be written in the form an (x) dx n + an−1 (x) dxn−1 +
dy
. . . + a1 (x) dx + a0 (x)y = f (x), where an (x), an−1 (x), . . . , a1 (x), a0 (x) are func-
tions of x and f (x) is a function of x.
Non-linear Differential Equations:
An ODE is non-linear if it cannot be written in the form mentioned above.
Differential Equations can also be classified based on their order:
First-Order Differential Equations:
An ODE is a first-order differential equation if the highest order of the
dy
derivative is 1. For example, dx + y = x is a first-order ODE.
Second-Order Differential Equations:
1
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