Differential Equations MATH C241

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Differential EquationsMATH C241
Class hours T Th S 2
(9.00 A.M. to 9.50 A.M.)

• Text Book Differential Equations
• with Applications and
• Historical Notes

by George F. Simmons
(Tata McGraw-Hill) (2003)
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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.

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Many important and significant problems in
engineering, the physical sciences, and the
social sciences, when formulated in mathematical
terms require the determination of a function
satisfying an equation containing derivatives of
the unknown functions. Such equations are called
differential equations i.e.
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A differential equation is a relationship between
an independent variable, (let us say x), a
dependent variable (let us call this y), and one
or more derivatives of y with respect to x.
is a differential equation.
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we recall that y f(x) is a given function, then
its derivatives dy/dx can be interpreted as the
rate of change of y with respect to x. In any
natural process, the variables involved and their
rates of change are connected with one another 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.
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Perhaps the most familiar example is Newtons law
For the position x(t) of a particle acted on by a
force F. In general F will be a function of time
t, the position x, and the velocity dx/dt.
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To determine the motion of a particle acted on by
a given force F it is necessary to find a
function x(t) satisfy the above equations. If the
force is that due to gravity, the F - mg and
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For example, the distance s traveled in time t by
a freely falling body of mass m satisfies the DE
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 is
Where k is the constant of proportionality 
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• The other examples of Physical phenomena
  involving rates of change are
• Motion of fluids
• Motion of mechanical systems
• Flow of current in electrical circuits

A DE that describes a physical process is often
called a Mathematical Model.
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Ordinary Differential Equations An ordinary
differential equation (ODE) is a differential
equation that involves the (ordinary) derivatives
or differentials of only a single independent
variable. equations are ODEs, while is not
ODE.
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In fact, the above equation is a partial
differential equation. A partial differential
equation (PDF) is a differential equation that
involves the partial derivatives of two or more
independent variables.
Heat equation
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Order

• The order of a differential equation is just the
  order of highest derivative used.

2nd order
.
3rd order
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Degree of a Differential Equation

• The power of the highest order derivative
  occurring in a differential equation, after it is
  free from radicals and fractions, is called the
  degree of a differential equation.

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Example The equation
is of second order and the second degree as the
equation can be written as
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More generally, the equation
(1)
is an ODE of the nth order. Equation (1)
represents a relation between the n2 variables
x, y, y, y, ., y(n) which under suitable
conditions can be solved for y(n) in terms of the
other variables
(2)
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Initial-Value Problem A differential equation
along with subsidiary conditions on the unknown
function and its derivatives, all given at the
same value of the independent variable,
constitutes an initial-value problem and the
conditions are initial conditions.
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For example The problem y'' 2y' ex y(p)
1, y'(p) 2 is an initial-value
problem,because the two subsidiary conditions are
both given at x p
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Boundary-Value problem If the subsidiary
conditions are given at more than one value of
the independent variable, the problem is a
boundary-value problem and the conditions are
boundary conditions.
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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 and x 1.
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A DE is said to be linear when the dependent
variable and all the derivatives of it appear
only in the 1st degree.
Examples
linear
linear
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Examples (Continued)
linear
Non linear
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Solutions of ODEs
A solution of an ODE
on the interval a, b is a function f such that
f, f, f, .f(n) exist for all x?a, b and
for all x?a, b.
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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
DE
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is a solution of the DE
is a solution of the DE
(Note here t is the independent variable and x ia
function of t.)
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General and particular solution
If no initial conditions are given, we call the
description of all solutions to the differential
equation the general solution.
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It is clear that the general solution of the DE
is the one-parameter family of parabolas
c is an arbitrary constant. (See the figure in
the next slide.)
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Figure 1 Graphs of
for various
value of C
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It can be shown that the general solution of
the DE
is the two-parameter family of curves
where c1, c2 are arbitrary constants.
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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
Differentiating w.r.t. x, we get
Eliminating c, we get the DE of the family as
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Find the DE of the family of all circles tangent
to the y-axis at the origin
Solution The equation to the circle tangent to
y-axis at the origin is given by
a
C
or
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Differentiating w.r.t. x, we get
or
Eliminating a, we get the DE of the family as
or
i.e.
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Consider the two-parameter family of curves
Differentiating w.r.t. x, we get
Again differentiating w.r.t. x, we get
Eliminating a,b, we get the DE of the family as
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Example Consider the DE
The general solution is
where c is an arbitrary constant.
We now show that there is a unique solution such
that when x 1, y 3.
Replacing x by 1, y by 3, we get a unique c,
namely, c 3.
Thus the desired unique solution is
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We now state (without proof) a theorem which
asserts that under suitable conditions that a
first order DE
has a unique solution y g(x) satisfying the
initial conditions when x x0, y y0
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Existence and uniqueness of solution of a first
order initial-value problem
Picards Theorem Consider the first order d.e.
Suppose
and
are both
continuous (as functions of x, y) at each point
(x, y) on and inside a closed rectangle R of the
x-y plane. Then for each point
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(x0, y0) inside the rectangle R, there exists a
unique solution y g(x) of the above DE such
that when x x0, y y0. Geometrically
speaking, through each point (x0, y0) inside the
rectangle R, there passes a unique solution curve
y g(x) of the DE
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yg(x)
(x0, y0)
R
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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 a
family of orthogonal trajectories of the other
family.
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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 easily see that
each is the family of orthogonal trajectories of
the other.
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If the DE of a one-parameter family ? of curves
in the xy plane is given by
from definition, it trivially follows that the DE
of the family ? of orthogonal trajectories is
given by
Integrating the above DE we get the algebraic
equation of the family of orthogonal
trajectories.
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Example Consider the one parameter family of
parabolas having the focus at the origin
c gt 0
c lt 0
The DE of the above family is
()
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Hence the DE of the family of orthogonal
trajectories is got by replacing
And hence is given by
or
which is same as ().
Hence the family of orthogonal trajectories is
the given family of parabolas itself. Or we say
that the given family of parabolas is
self-orthogonal.
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In the next lecture we discuss the methods of
solving first order differential equations.