Differential Equations

2nd Lecture

Instructor: Ahmed Salah Jamal

ahmed.salah@tiu.edu.iq

Tishk International University

Department of Civil Engineering

Fall - 2021

Introduction to Differential

Equations

Course Textbook

2

Zill, D. G. (2018). Advanced engineering mathematics (6th ed.).

Jones & Bartlett Learning, Burlington, Massachusetts, USA.

Grading Criteria

3

• Class Activity and Attendance 10%

• Assignments 10%

• Quizzes 15%

• Midterm Exam. 25%

• Final Exam. 40%

Attention

4

Students should have basic knowledge about:

- Functions and their domains and ranges.

- Differentiation (of nearly all types of functions).

- Different methods of integration.

- Understanding physical problems, the relationships between the variable, and how to translate these relations to

mathematical models.

An equation containing the derivatives of one or more dependent variables, with respect to one or more

independent variables, is said to be a differential equation (DE).

Differential Equations

DE Classification by Type

Ordinary Differential Equations (ODE) Partial Differential Equations (PDE)

An equation containing only ordinary derivatives of

one or more functions with respect to a single

independent variable.

An equation involving only partial derivatives of one

or more functions of two or more independent

variables

Examples:

Examples:

The order of the differential equation (ODE or PDE) is the order of the highest derivative in the equation.The degree of a differential equation is the power (exponent) of the highest order derivative term in the differential equation.

First-order ODE

Second-order ODE

Third-order ODE

Second-order ODE

Fourth-order PDE

DE Classification by Order

Differential Equation Order of DE Degree of DE

32 xdx

dy

36

4

3

3

y

dx

dy

dx

yd

First

Third

First

First

First

03

53

2

2

dx

dy

dx

yd

DE Classification by Linearity

Linear Differential Equations Non-Linear Differential Equations

If the dependent variable y and all its derivatives y’, y’’,

…., y(n) are of the first degree; i.e. the power of each

term involving y is 1 in the DE, then the DE is linear.

Linear differential equation:

1st - order ODE: 𝒂 𝒙 𝐲′ + b 𝒙 𝒚= c 𝒙

2nd - order ODE: 𝒂 𝒙 𝐲′′ + b 𝒙 𝒚′ + c 𝒙 𝐲 = d 𝒙

If the coefficients of y, y’, …., y(n) contain the

dependent variable y or its derivatives or if powers of

y, y’, …., y(n), such as (y’)2, appear in the equation,

then the DE is nonlinear. Also, nonlinear functions of

the dependent variable or its derivatives, such as sin y

or 𝑒𝑦 cannot appear in a linear equation.

Non-Linear differential equation:

1st - order ODE: 𝒚𝐲′ = ‐ 𝒙

2nd - order ODE: (𝐲′′)2 = 𝒚𝒙

Examples:

Examples:

Explicit and Implicit Solutions of Differential Equations

Explicit Solution Implicit Solution

An explicit solution is any solution that is given in the

form y=y(t). In other words, the only place that y

actually shows up is once on the left side and only

raised to the first power.

Let's say that y is the dependent variable and x is the

independent variable. An explicit solution would be

y=f(x), i.e. y is expressed in terms of x only.

An implicit solution is any solution that isn’t in explicit

form.

An implicit solution is when you have f(x,y)=g(x,y)

which means that y and x are mixed together. y is not

expressed in terms of x only. You can have x and y

on both sides of the equal sign or you can have y on

one side and x,y on the other side. An example of

implicit solution is 𝑦 = 𝑥 𝑥 + 𝑦 2

Examples:

Examples:

9

Initial Value Problems

An Initial Value Problem (or IVP) is a differential equation along with an appropriate number of initialconditions that specify the value of the unknown function at a given point in the domain.

forSolve

forSolve

Note that the number of initial conditions required will depend on the order of the differential equation.

Examples:

Solve for

Solve for

General and Particular Solutions of Differential Equations

General Solution

A function which satisfies the given differential equation is called its solution. The general solution to a differential equation is

the most general form that the solution can take and it doesn’t take any initial conditions into account. It should be noted that

in the general solution, the number of arbitrary constants equal to the order of the DE.

1st Order one arbitrary constant

2nd Order two arbitrary constants

3rd Order three arbitrary constants

Examples:

The particular (actual) solution to a differential equation is the specific solution that not only satisfies the differential equation,

but also satisfies the given initial condition(s). That means that it can be obtained from general solution by given particular

values to the arbitrary constants.

Solve for

Solve for

Particular Solution

11