Transcript differential equation

Modeling with differential equations

One of the most important application of calculus
is differential equations, which often arise in
describing some phenomenon in engineering,
physical science and social science as well.
Concepts of differential equations
In general, a differential equation is an equation that
contains an unknown function and its derivatives. The order
of a differential equation is the order of the highest derivative
that occurs in the equation.
 A function y=f(x) is called a solution of a differential
equation if the equation is satisfied when y=f(x) and its
derivatives are substituted into the equation.

Example
Ex. Show that every member of the family of functions
1  cet
y
,
t
1  ce
1 2
where c is an arbitrary constant, is a solution of y  ( y  1).
2
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Sol.
(1  cet )cet  (1  cet )(cet )
2cet
y 

t 2
(1  ce )
(1  cet )2
1 2
1  (1  cet )2 
2cet
1 2

 y  ( y  1).
( y  1)  
 1 
t 2
t 2
2
2
2  (1  ce )
 (1  ce )
Concepts of differential equations
If no additional conditions, the solution of a differential
equation always contains some constants. The solution family
that contains arbitrary constants is called the general solution.
 In real applications, some additional conditions are
imposed to uniquely determine the solution. The conditions
are often taken the form y (t0 )  y0 , that is, giving the value
of the unknown function at the end point. This kind of
condition is called an initial condition, and the problem of
finding a solution that satisfies the initial condition is called
an initial-value problem.
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Geometric point of view
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Geometrically, the general solution is a family of solution
curves, which are called integral curves.
When we impose an initial condition, we look at the family
of solution curves and pick the one that passes through the
point (t0 , y0 ).
Physically, this corresponds to measuring the state of a
system at time t 0 and using the solution of the initial-value
problem to predict the future behavior of the system.
Example

Ex. Solve the initial-value problem
  1 2
 y  ( y  1)
.
2

 y(0)  2
1  cet
 Sol. Since the general solution is y 
, substituting
t
1  ce
the values t=0 and y=2, we have
1  ce0 1  c
1
2


c

.
0
1  ce 1  c
3
1  13 et 3  et
.
So the solution of the initial-value problem is y  1 t 
t
1 3 e 3  e
Graphical approach: direction fields
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For most differential equations, it is impossible to find an
explicit formula for the solution.
Suppose we are asked to sketch the graph of the solution of
the initial-value problem y  f ( x, y ), y ( x0 )  y0 .
The equation tells us that the slope at any point (x,y) on the
graph is f(x,y).
To sketch the solution curve, we draw short line segments
with slope f(x,y) at a number of points (x,y). The result is
called a direction field.
Example

Ex. Draw a direction for the equation y  4  2 y. What can
you say about the limiting value when x .
Sol.
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Remark: equilibrium solution
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Separable equations
Not all equations have an explicit formula for a solution.
But some types of equations can be solved explicitly.
Among others, separable equations is one type.
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A separable equation is a first-order differential equation in
which the expression for dy / dx can be factored into the
product of a function of x and a function of y. That is, a
separable equation can be written in the form
dy
 f ( x) g ( y ).
dx
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Solutions of separable equations
dy
 Thus a separable equation can be written into
 f ( x)dx,
g ( y)
that is, the variables x and y are separated!
 We can then integrate both sides to get:
dy
 g ( y)   f ( x)dx.
After we find the indefinite integrals, we get a relationship
between x and y, in which there generally has an arbitrary
constant. So the relationship determines a function y=y(x) and
it is the general solution to the differential equation.
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Example
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Ex. Solve the differential equation y  (1  y 2 ) x 2 .

dy
2

x
dx.
Sol. Rewrite the equation into
2
1 y
dy
2
Integrate both sides 

x
dx,
2

1 y
which gives
1 3
arctan y  x  C.
3
1 3
So the general solution is y  tan( x  C ).
3
Example
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6 x2
.
Ex. Solve the differential equation y 
2 y  cos y
Sol. Separate variables: (2 y  cos y)dy  6 x 2 dx
Integrate:
y 2  sin y  2 x3  C
which is the general solution in implicit form.
Remark: it is impossible to solve y in terms of x explicitly.
Example
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Ex. Solve the differential equation y  x 2 y.
3
x3
3
dy
x
2
C

x
dx

ln
|
y
|


C

y


e
e
 Sol.
y
3
C is arbitrary, but  eC is not arbitrary. While we can verify
y=0 is also a solution. Therefore
y  Ae
x3
3
where A is an arbitrary constant, is the general solution.
Orthogonal trajectories
An orthogonal trajectory of a family of curve is a curve
that intersects each curve of the family orthogonally. For
instance, each member of the family y  mx of straight lines
is an orthogonal trajectory of the family x 2  y 2  r 2 .
 To find orthogonal trajectories of a family of curve, first
find the slope at any point on the family of curve, which is
generally a differential equation. At any point on the
orthogonal trajectories, the slope must be the negative
reciprocal of the aforementioned slope. So the slope of
orthogonal trajectories is governed by a differential equation,
too. Last solve the equation to get the orthogonal trajectories.
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Example
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Ex. Find the orthogonal trajectories of the family of curves
x  ky 2 , where k is an arbitrary constant.
dy
1
2

Sol. Differentiating x  ky , we get dx  2kydy, or
dx 2ky
Substituting k  x / y 2 into it, we find the slope at any point is
dy y
 .
dx 2 x
dy
2x
 .
At any point on orthogonal trajectory, the slope is
dx
y
2
Solving the equation, we get 2 y
x   C.
2
Example
2x
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Ex. Suppose f is continuous and f ( x)  
0
Find f(x).
x
Sol.  f ( x)  2 f (u)du  ln 2
0
 f ( x)  2 f ( x), f (0)  ln 2
 f ( x)  e2 x ln 2.
t
f ( )dt  ln 2.
2
Homework 21
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Section 8.2: 8, 14, 29
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Section 8.3: 28, 29
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Page 583: 7, 8, 10
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Section 9.1: 10, 11