01 introduction - copy
TRANSCRIPT
-
7/30/2019 01 Introduction - Copy
1/43
Differential Equations
MATH C241
Text Book: Differential Equations
with Applications and
Historical Notes:
Class hours: T Th S 2
(9.00 A.M. to 9.50 A.M.)
by George F. Simmons
(Tata McGraw-Hill) (2003)
-
7/30/2019 01 Introduction - Copy
2/43
In this introductory lecture, we
Define a differential equation Explain why we study a differential equation
Define the order and degree of a DE.
Define the solution of a DE
Formation of a DE
Discuss the Orthogonal trajectories of a family of
curves.
-
7/30/2019 01 Introduction - Copy
3/43
Many important and significant
problems in engineering, the
physical sciences, and the social
sciences, when formulated in
mathematical terms require thedetermination of a function satisfying
an equation containing derivatives of
the unknown functions. Suchequations are called differential
equations i.e.
-
7/30/2019 01 Introduction - Copy
4/43
A dif ferent ial equation is a
relationship between an independentvariable, (let us say x), a dependent
variable (let us call this y), and one or
more derivatives of y with respect tox. 2
3
2 0xd y dy
x y edxdx
is a differential equation.
-
7/30/2019 01 Introduction - Copy
5/43
we recall that y= f(x) is a given function,
then its derivatives dy/dx can be
interpreted as the rate of change of ywith respect to x. In any natural
process, the variables involved and their
rates of change are connected with oneanother by means of the basic scientific
principles that govern the process.When this connection is expressed in
mathematical symbols, the result is
often a differential equation.
-
7/30/2019 01 Introduction - Copy
6/43
Perhaps the most familiar example
is Newtons law2
2
d xm F
dt
For the position x(t) of a particle acted
on by a force F. In general F will be afunction of time t, the position x, and the
velocity dx/dt.
-
7/30/2019 01 Introduction - Copy
7/43
To determine the motion of a particle
acted on by a given force F it is
necessary to find a function x(t) satisfythe above equations. If the force is that
due to gravity, the F = - mg and
2
2
d xm mg
dt
-
7/30/2019 01 Introduction - Copy
8/43
For example, the distances traveled in time t
by a freely falling body of mass m satisfies
the DE2
2
d sg
dt
The time rate of change of a population P(t) with
constant birth and death rates is, in many simple cases,
proportional to the size of the population. That isdP
kPdt
Where k is the constant of proportionality
-
7/30/2019 01 Introduction - Copy
9/43
The other examples of Physical
phenomena involving rates of changeare :
Motion of fluids Motion of mechanical systems
Flow of current in electrical circuitsA DE that describes a physical process is
often called a Mathematical Model.
-
7/30/2019 01 Introduction - Copy
10/43
Ordinary Differential Equations:
An ordinary differential equation (ODE) isa differential equation that involves the
(ordinary) derivatives or differentials of
only a single independent variable.
equations are ODEs, while
is not ODE.
y'' - 2y' + y = cos
sinx
x
dye xdx
2 2
2 2
u u u
tx y
-
7/30/2019 01 Introduction - Copy
11/43
In fact, the above equation is a
partial differential equation. A partialdifferential equation (PDF) is adifferential equation that involves the
partial derivatives of two or moreindependent variables.
22
2w wa
tx
Heat equation
-
7/30/2019 01 Introduction - Copy
12/43
Order
The order of a differential equation isjust the order of highest derivativeused.
02
2
dt
dy
dt
yd
.2nd order
3
3
dt
xdx
dt
dx 3rd order
-
7/30/2019 01 Introduction - Copy
13/43
Degree of a Differential Equation
The power of the highest order
derivative occurring in a differential
equation, after it is free fromradicals and fractions, is called the
degree of a differential equation.
-
7/30/2019 01 Introduction - Copy
14/43
2 3 2 2 2 2{1 ( ) } ( / )dy
a d y dxdx
Example: The equation
22 3 / 2
2{1 ( ) }
dy d ya
dx dx
is of second order and the second degree
as the equation can be written as
-
7/30/2019 01 Introduction - Copy
15/43
More generally, the equation
( )( , , , , ) 0nF x y y y is an ODE of the nth order. Equation (1)
represents a relation between the n+2
variables x, y, y, y, ., y(n) which under
suitable conditions can be solved for y
(n)
in terms of the other variables:
(1)
( ) ( 1)
( )
n n
y f x y y y
(2)
-
7/30/2019 01 Introduction - Copy
16/43
Initial-Value Problem:
A differential equation along with
subsidiary conditions on the unknown
function and its derivatives, all givenat the same value of the independent
variable, constitutes an initial-value
problem and the conditions are initial
conditions.
-
7/30/2019 01 Introduction - Copy
17/43
For example: The problem
y'' + 2y' = ex; y(p) = 1, y'(p) = 2
is an initial-value problem,because
the two subsidiary conditions areboth given at x = p
-
7/30/2019 01 Introduction - Copy
18/43
Boundary-Value problem:
If the subsidiary conditions are given
at more than one value of the
independent variable, the problem isa boundary-value problem and the
conditions are boundary conditions.
-
7/30/2019 01 Introduction - Copy
19/43
For Example:The problem
y'' + 2y' = ex; y(0) =1, y'(1) = 1
is a boundary-value problem, because
the two subsidiary conditions are
given at the different values x = 0 andx = 1.
-
7/30/2019 01 Introduction - Copy
20/43
A DE is said to be linear when the
dependent variable and all the derivativesof it appear only in the 1st degree.
Examples
x
dx
dy2.1
ydx
yd
2
2
.2
linear
linear
-
7/30/2019 01 Introduction - Copy
21/43
xeydxdy
dxyd 32
2
65.3
Examples (Continued)
2 24. 1 0y y y
linear
Non linear
-
7/30/2019 01 Introduction - Copy
22/43
Solutions of ODEs:
A solution of an ODE( )( , , , , ) 0 (1)nF x y y y
on the interval [a, b] is a function f such that
f, f, f, .f(n) exist for all x[a, b] and
( ), ( ), ( ), ( ) 0nF x f x f x f x
for all x[a, b].
-
7/30/2019 01 Introduction - Copy
23/43
Given a DE any relation between the
variables (that is free from derivatives) that
satisfies the DE is called the solution of the
DE
For example is a solution of the DE2y x
2dy xdx
-
7/30/2019 01 Introduction - Copy
24/43
2 2 4x y is a solution of the DE
0xdx y dy sin2x t is a solution of the DE
2
24
d xx
dt
(Note here tis the independent variable and
x ia function oft.)
-
7/30/2019 01 Introduction - Copy
25/43
If no initial conditions are given, we call
the description of all solutions to the
differential equation the general solution.
General and particular solution:
cos siny' x y x c general solution
sin 2 siny x or y x particaular solution
-
7/30/2019 01 Introduction - Copy
26/43
It is clear that the general solution of the
DE 2dy xdx
is the one-parameter family of parabolas2y x c
c isan arbitrary constant.
(See the figure in the next slide.)
-
7/30/2019 01 Introduction - Copy
27/43
Figure 1 Graphs of 2y x C for various
value of C
-
7/30/2019 01 Introduction - Copy
28/43
It can be shown that the general solution
of the DE
is the two-parameter family of curves
1 2cos 2 sin 2x c t c t
where c1, c2are arbitrary constants.
2
24
d xx
dt
-
7/30/2019 01 Introduction - Copy
29/43
Conversely, given a family of curves, we can
find the DE satisfied by the family (by
eliminating the parameters by differentiation).
Consider the one-parameter family of curves
2y c xDifferentiating w.r.t. x, we get 2
dyc x
dx
Eliminating c, we get the DE of the family as
2dy
x y
dx
-
7/30/2019 01 Introduction - Copy
30/43
Find the DE of the family of all circles
tangent to they-axis at the origin
Ca
Solution The equation
to the circle tangent to
y-axis at the origin isgiven by
2 2 2( )x a y a
or2 22 0x ax y
-
7/30/2019 01 Introduction - Copy
31/43
or 0x a yy
Eliminating a, we get the DE of the family as2 2
02
x yx yy
x
or
2 2
2
y xy
xy
Differentiating w.r.t. x, we get
2 2 2 0x a yy
i.e.
2 2
0
2
x yyy
x
-
7/30/2019 01 Introduction - Copy
32/43
Consider the two-parameter family of curves2 2
2 2 1
x y
a b Differentiating w.r.t. x, we get
2 2
2 20
x y dy
a b dx
Eliminating a,b, we get the DE of the family as
Again differentiating w.r.t. x, we get22
2 2 2
2 20
d y dyy
a b dx dx
22
20
d y dy dyxy x y
dx dx dx
-
7/30/2019 01 Introduction - Copy
33/43
Example: Consider the DE xdy
xedx
The general solution isx xy xe e c
where c is an arbitrary constant.
We now show that there is a unique solutionsuch that whenx = 1,y =3.
Replacing x by 1, y by 3, we get a
unique c, namely, c = 3.Thus the desired unique solution is
3
x x
y xe e
-
7/30/2019 01 Introduction - Copy
34/43
We now state (without proof) a theorem
which asserts that under suitable conditions
that a first order DE
( , )dy
f x y
dx
has a unique solutiony =g(x) satisfying the
initial conditions: whenx = x0,y =y0
-
7/30/2019 01 Introduction - Copy
35/43
Existence and uniqueness of solution of a
first order initial-value problem
Picards Theorem
Consider the first order d.e. ( , )dy
f x ydx
Suppose ( , )f x y andf
y
are both
continuous (as functions ofx, y) at each
point (x,y) on and inside a closed rectangle
R of the x-y plane. Then for each point
-
7/30/2019 01 Introduction - Copy
36/43
(x0,y0) inside the rectangle R, there exists a
unique solution y = g(x) of the above DE
such that whenx =x0,y =y0.
Geometrically speaking, through each point
(x0,y0) inside the rectangle R, there passes aunique solution curvey =g(x)of the DE
( , )dy
f x ydx
-
7/30/2019 01 Introduction - Copy
37/43
(x0,y0)
y=g(x)
R
-
7/30/2019 01 Introduction - Copy
38/43
Orthogonal trajectories of a family of
curves
Consider two families of curves, , in the
xy plane. Suppose every curve in the family
intersects every curve in the family
orthogonally (i.e. the angle between the two
curves at each point of intersection is 90o, i.e.
a right angle), then each family is said to be afamily of orthogonal trajectories of the other
family.
-
7/30/2019 01 Introduction - Copy
39/43
For example if is the family of all cirles
centre at the origin and is the family of
all lines through the origin, then we easilysee that each is the family of orthogonal
trajectories of the other.
-
7/30/2019 01 Introduction - Copy
40/43
If the DE of a one-parameter family of
curves in the xy plane is given by
( , )dy f x ydx
from definition, it trivially follows that the DEof the family of orthogonal trajectories is
given by1
( , )
dy
dx f x y
Integrating the above DE we get the algebraic
equation of the family of orthogonal trajectories.
-
7/30/2019 01 Introduction - Copy
41/43
Example Consider the one parameter family
of parabolas having the focus at the origin:
2 4 ( )y c x c
The DE of the above family is:
2 2
y yyx
y
(*)
c > 0 c < 0
-
7/30/2019 01 Introduction - Copy
42/43
Hence the DE of the family of orthogonal
trajectories is got by replacing
And hence is given by
2 2
yy yx
y
1y by
y
2 2
y yyx
y
or
which is same as (*).
Hence the family of orthogonal trajectories isthe given family of parabolas itself. Or we say
that the given family of parabolas is self-
orthogonal.
-
7/30/2019 01 Introduction - Copy
43/43
In the next lecture we discuss the
methods of solving first orderdifferential equations.