1. Introduction to Differential Equations

Engineering Mathematics I

Dr. Rami Zakaria

Engineering Mathematics I_ 2017

Terminology Nomenclature

Differential Equation DE

Ordinary Differential Equations ODE

Partial Differential Equations PDE

Initial-Value Problem IVP

Ref [1] Advanced Engineering Mathematics, 4th ed., by Dennis G. Zill & Warren S. Wright

Part 1:

A review of the concept of

differentiation

2

3

Suppose we have a variable y that is a function of another variable

x, such as y = f (x).

the rate Δy/ Δx explains how “fast” y changes for a certain shift in

the value of x.

For a simple function such as y=5x, if x changes from 1 to 4 (for

example), y moves from 5 to 20. Therefore, the rate Δy/ Δx = 5 …

always!

The concept of differentiation

4

Remember the definition of the derivative of a function:

it is not practical to apply the previous relation every time we se

ek to calculate the derivative of a function. This is why we gath

ered rules that we can apply directly according to the type of the

function (see the attached table “DERIVATIVES AND INTEGRALS”).

Note: in this class we will use the (prime notion):

, and the derivative shape:

n

n

n

dx

yd

dx

yd

dx

yd

dx

dy

yyyyy

,...,,,

,...,,,,

3

3

2

2

)()4(

(… Although we will avoid using the dot notion, you should be aware of it)

dx

dy

x

y

x

xfxxfxf

xx

00lim

)()(lim)(

5

The first derivative f ’ of a curve f at a certain

point is the tangent line on the curve at that point.

Lmxf )( 0

The graphical meaning of differentiation

The first derivative f’ can be estimated from the

slope of the tangent line at each point of f.

6

When we write y = f (x), we are implying that y reacts to t

he changes of x. In other words, x changes first then y foll

ows. In this case, we call x an independent variable, and

y a dependent variable.

If we have T(x,t) or T(t,x), then T is dependent on both t a

nd x.

In models, we can understand the independent variables

as the inputs, while the dependent variables as the output.

Dependent & Independent Variables

7

Ordinary and Partial Derivatives

Let f (x) be a function dependent only on x. We can then

calculate the ordinary derivative

If we have g(x,t) a function dependent on both x and t, then

it is possible to calculate the partial derivatives:

When we calculate a partial derivative we assume that one

independent variable is changing while the others are

constants.

)(xfdx

d

),(and),( txft

txfx

8

?),(

)!Derivative (Total DerivativeOrdinary

2),(Derivative Partial

),(Derivative Partial

2)(DerivativeOrdinary

2)(DerivativeOrdinary

),(,)(,1)(

2

2

222

dt

yxdf

xyyxfy

yyxfx

etydt

d

ttxdt

d

xyyxfetyttx

t

t

Example - we have the following functions:

Part 2: Definition and Classification of A

Differential Equation

9

A differential equation (DE) is an equation that contains the derivatives of one or more dependent variables, with respect to one or more independent variables.

A derivative of a function f(x) is another function f’(x)=dy/dx, that can be found using the general differentiation rules. For example :

If [2] is my equation, then I can say that [1] is a solution for that equation. 10

xx exyexy 653 2

xexdx

dy 6Or:

1

2

22 xdx

dyFor example: is a differential equation where { x }is the independent variable and

{ y(x) } is the dependent variable

What is a differential equation?

Another example;

11

xyy

Then can you tell if 25.04 xey is a solution for it?

If I have this differential equation:

The answer is yes! It is a solution.

In fact, this is not the only solution.

Try for example you will find it is also a solution. 25.06 xey

But the big question is:

HOW CAN I FIND THESE SOLUTIONS?

We will answer this question during this course, but before that, we need to

classify differential equations by type, order, and linearity.

12

ODE : Ordinary Differential Equation

Contains derivates of only one

independent variable.

Examples:

PDE : Partial Differential Equation

Contains derivates of more than one

independent variable.

Examples:

yedt

xd

dt

dy

xydx

dy

x

2

2

3

sin2

03

32

2

t

u

y

u

x

u

yt

y

x

y

Type

The order of a differential

equation is the order of the

highest derivative.

Examples:

xeyyx

ydx

dy

dx

yd

02 2

4

3

3

3

order

2

Order

The degree of a differential equation is power

of the highest order derivative term.

Example:

032

2

aydx

dy

dx

yd

03

53

2

2

dx

dy

dx

yd

degree

3

1

Degree

Linear

Or

Non-linear

Linearity

Classification of Differential Equations

Linear Differential Equation

A differential equation is linear, if :

1. dependent variable and its derivatives are of degree one,

2. coefficients of a term do not depend upon dependent variable.

36

4

3

3

y

dx

dy

dx

yd is non - linear because in 2nd term is not of degree one.

.0932

2

ydx

dy

dx

yd

Examples:

is linear. 1.

2.

3

2

22 x

dx

dyy

dx

ydx 3. is non - linear because in 2nd term coefficient depends on y.

13

nth – order linear differential equation

1. nth – order linear differential equation with constant coefficients.

xgyadx

dya

dx

yda

dx

yda

dx

yda

n

n

nn

n

n

012

2

21

1

1 ....

2. nth – order linear differential equation with variable coefficients

xgyxadx

dyxa

dx

ydxa

dx

ydxa

dx

dyxa

n

n

nn

012

2

2

1

1 ......

14

15

A function ϕ(x) is a solution of an ODE on an interval (domain) I,

if it satisfies the ODE on I.

For Example : y=3x+c1

is a solution of a 1st order ODE

On the interval (-∞,∞)

3dx

dy

Exercise (P5) :

Solution:

Verify that : 4

16

1xy is a solution of the DE:

5.0xyy On the interval (-∞,∞)

Left-hand side:

right-hand side:

4164

33 xxy

416

35.0

45.0 xx

xxy

Notice in this example y=0 is one of

the solutions. In DEs we call the zero

solution a trivial solution.

Solution of an ODE

Explicit and Implicit solutions

16

An Explicit Solution is a solution where the dependent

variable y is expressed in terms of the independent variable

x and constants. In other words, it is a solution written as

y=ϕ(x).

Sometimes, when we solve a DE, we don’t directly get an

explicit solution. Instead we find a function G(x,y)=0 that

satisfies the equation; we call it an Implicit Solution.

Example : 2522 yx is an implicit solution of the differential equation y

x

dx

dy

In fact any function cyx 22 would satisfy the equation; where c is an arbitrary constant

17

Notice that the solution is a family of ellipses.

Observe that at any given point (x0,y0), there is a particular solution

(unique) curve of the above equation which goes through the given point.

049 xyycxy

94

22

Another example: is a solution for the DE:

We call this type of solution a Family of solutions or a General Solution*

Remember that an equation can also have a singular solution that is not a member of a family!

A System of Differential Equations:

Until now we are discussing one differential equation. But in many cases we will need to solve two or

more equations that share the same independent variable together.

For example, when we look for a solution of this system of equations:

),,(

),,(

yxtgdt

dy

yxtfdt

dx

we are actually looking for solutions x=ϕ1(t) and y= ϕ2(t) that satisfy the two equations at the same time.

* Usually we use the term General Solution if the family of solutions are the only solutions of the equation (which is true in many cases).

Exercises:

18

Equation Order Linearity

2 Non-linear 02

3

2

22

y

dx

dy

dx

ydx

3)4(2 3tuuut

xeyyxyxyx 4sin32

2

2

yy

22 xxyyy

xyyex 1

3

02 yyy

1. Determine the order and linearity of the following ODE:

19

5. Verify that : xx xececy 2

2

2

1 is a general solution (family of solutions) for the ODE:

044 yyy

6. Verify that :

0

02

2

xx

xxy is a solution for : 02 yyx on the interval (-∞,∞)

See more examples at P.10

20

2. Verify that : 2/xey is a solution of the ODE: 02 yy

3. Verify that : xey x 2cos3 is a solution of the ODE: 0136 yyy

See more examples at P.9

4. Verify that : 12 22 yyx is an implicit solution of the ODE: 0)(2 2 dyyxxydx

and then find the explicit solution(s).

21

22

6. Determine if : 935 yyx has (accepts) a constant solution or not.

3393

0

ycc

ycyAssume:

Then: is a solution

7. Determine if : 11 yy has (accepts) a constant solution or not.

8. Determine if pair of functions: tttt eeyeex 6262 5,3

is a solution for the system of differential equations: yxdt

dyyx

dt

dx35,3