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Math 230 Sec 1.1

This document defines key terms related to differential equations including: - Ordinary differential equations contain derivatives of dependent variables with respect to a single independent variable, while partial differential equations contain derivatives with respect to two or more independent variables. - The order of a differential equation is the highest derivative that appears. Linear differential equations are additive, while nonlinear equations are not additive. - A function is a solution to a differential equation if it satisfies the equation. Solutions can be trivial, explicit, or implicit. Differential equations often have families of solutions parameterized by arbitrary constants.

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156 views5 pages

Math 230 Sec 1.1

This document defines key terms related to differential equations including: - Ordinary differential equations contain derivatives of dependent variables with respect to a single independent variable, while partial differential equations contain derivatives with respect to two or more independent variables. - The order of a differential equation is the highest derivative that appears. Linear differential equations are additive, while nonlinear equations are not additive. - A function is a solution to a differential equation if it satisfies the equation. Solutions can be trivial, explicit, or implicit. Differential equations often have families of solutions parameterized by arbitrary constants.

Uploaded by

Trav Black
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Math 230 – Section 1.

1 Basic definitions and terminology

I. Differential equations

Definition: A differential equation (DE) is an equation containing the derivatives or


differentials of one or more dependent variables, with respect to one or more independent
variables.

Ex: Newton’s equation

𝑑𝑑2 𝑥𝑥
𝑚𝑚 = 𝐹𝐹(𝑡𝑡, 𝑥𝑥)
𝑑𝑑𝑑𝑑 2
Classification: by type, order, and linearity

Type:

1. Ordinary differential equation (ODE): contains only ordinary derivatives of one or more
dependent variables with respect to a single independent variable.

Ex:

2. Partial differential equation (PDE): contains partial derivatives of one or more dependent
variables with respect to 2 or more independent variables.

Ex:

a) Laplace equation

b) Heat equation

c) Wave equation

1
Order:

The order of a DE is the highest derivative’s order that appears in the equation.

Ex:

The general form for the n-th order ODE is

𝑑𝑑𝑑𝑑 𝑑𝑑2 𝑦𝑦 𝑑𝑑𝑛𝑛 𝑦𝑦


𝐹𝐹 �𝑥𝑥, 𝑦𝑦, , 2 , ⋯ , 𝑛𝑛 � = 0
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

Linearity:

1. Linear:

Ex:

The general form for a linear ODE is

𝑑𝑑 𝑛𝑛 𝑦𝑦 𝑑𝑑 𝑛𝑛−1 𝑦𝑦 𝑑𝑑2 𝑦𝑦 𝑑𝑑𝑑𝑑


𝑎𝑎𝑛𝑛 (𝑥𝑥) 𝑛𝑛 + 𝑎𝑎𝑛𝑛−1 (𝑥𝑥) 𝑛𝑛−1 + ⋯ + 𝑎𝑎2 (𝑥𝑥) 2 + 𝑎𝑎1 (𝑥𝑥) + 𝑎𝑎0 (𝑥𝑥)𝑦𝑦 = 𝑔𝑔(𝑥𝑥)
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

2. Nonlinear:

Ex:

2
II. Solution of a differential equation

Definition: A function 𝑓𝑓 defined on an interval 𝐼𝐼 is said to be a solution of a DE if it reduces the


equation to an identity when substituted into the equation.

Ex: verify that the indicated function is a solution of the given differential equation.

𝑑𝑑𝑑𝑑 6 6
a) + 20𝑦𝑦 = 24; 𝑦𝑦 = − 𝑒𝑒 −20𝑥𝑥
𝑑𝑑𝑑𝑑 5 5

𝜋𝜋
b) 𝑦𝑦 ′′ + 𝑦𝑦 = sec 𝑡𝑡 ; 𝑦𝑦 = cos 𝑡𝑡 ln(cos 𝑡𝑡) + 𝑡𝑡 sin 𝑡𝑡 for 0 < 𝑡𝑡 <
2

3
c) 2𝑥𝑥𝑥𝑥 𝑑𝑑𝑑𝑑 + (𝑥𝑥 2 + 2𝑦𝑦)𝑑𝑑𝑑𝑑 = 0; 𝑥𝑥 2 𝑦𝑦 + 𝑦𝑦 2 = 𝑐𝑐 where 𝑐𝑐 is a constant.

Definitions:

1. Trivial solution:

2. Explicit solution:

3. Implicit solution:

n-parameter family of solutions:

Ex: one-parameter family of solutions

Consider the ODE 𝑦𝑦 ′ = 𝑥𝑥𝑦𝑦1/2

4
Ex: two-parameter family of solutions

Consider the ODE 𝑦𝑦 ′′ − 𝑦𝑦 = 0

Remark: In general, when solving an nth-order equation, we expect an n-parameter family of


solutions.

Definition: A solution that is free of arbitrary parameters is called a particular solution. A


solution that cannot be obtained by specializing the parameters is called a singular solution.

Ex: Consider the ODE 𝑥𝑥𝑦𝑦 ′ − 4𝑦𝑦 = 0

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