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