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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. - PowerPoint PPT Presentation
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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
where c is an arbitrary constant, is a solution of
Sol.
1,
1
t
t
cey
ce
21
( 1).2
y y
2 2
(1 ) (1 )( ) 2
(1 ) (1 )
t t t t t
t t
ce ce ce ce cey
ce ce
22
2 2
1 1 (1 ) 2( 1) 1
2 2 (1 ) (1 )
t t
t t
ce cey
ce ce
21( 1).
2y y
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 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.
0 0( ) ,y t y
Geometric point of view 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
Physically, this corresponds to measuring the state of a system at time and using the solution of the initial-value problem to predict the future behavior of the system.
0 0( , ).t y
0t
Example
Ex. Solve the initial-value problem
Sol. Since the general solution is substituting
the values t=0 and y=2, we have
So the solution of the initial-value problem is13
13
1 3.
1 3
t t
t t
e ey
e e
21( 1)
.2(0) 2
y y
y
1,
1
t
t
cey
ce
0
0
1 1 12 .
1 1 3
ce cc
ce c
Graphical approach: direction fields 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 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.
0 0( , ), ( ) .y f x y y x y
Example Ex. Draw a direction for the equation What
can you say about the limiting value when Sol.
Remark: equilibrium solution
4 2 .y y .x
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.
A separable equation is a first-order differential equation in
which the expression for 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 dx
( ) ( ).dy
f x g ydx
Solutions of separable equations Thus a separable equation can be written into
that is, the variables x and y are separated! We can then integrate both sides to get:
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.
( ) ,( )
dyf x dx
g y
( ) .( )
dyf x dx
g y
Example Ex. Solve the differential equation
Sol. Rewrite the equation into
Integrate both sides
which gives
So the general solution is
2 2(1 ) .y y x
22
.1
dyx dx
y
22
,1
dyx dx
y
31arctan .
3y x C
31tan( ).
3y x C
Example Ex. Solve the differential equation
Sol. Separate variables:
Integrate:
which is the general solution in implicit form. Remark: it is impossible to solve y in terms of x explicitly.
26.
2 cos
xy
y y
2(2 cos ) 6y y dy x dx 2 3sin 2y y x C
Example Ex. Solve the differential equation
Sol.
C is arbitrary, but is not arbitrary. While we can verify
y=0 is also a solution. Therefore
where A is an arbitrary constant, is the general solution.
2 .y x y33
2 3ln | |3
xCdy x
x dx y C y e ey
Ce
3
3
x
y Ae
Orthogonal trajectories An orthogonal trajectory of a family of curve is a curvethat intersects each curve of the family orthogonally. Forinstance, each member of the family of straight linesis an orthogonal trajectory of the family To find orthogonal trajectories of a family of curve, firstfind the slope at any point on the family of curve, which isgenerally a differential equation. At any point on theorthogonal trajectories, the slope must be the negativereciprocal of the aforementioned slope. So the slope oforthogonal trajectories is governed by a differential equation,too. Last solve the equation to get the orthogonal trajectories.
y mx2 2 2.x y r
Example Ex. Find the orthogonal trajectories of the family of curves
where k is an arbitrary constant. Sol. Differentiating we get or
Substituting into it, we find the slope at any point is
At any point on orthogonal trajectory, the slope is
Solving the equation, we get
2 ,x ky2 ,x ky 2 ,dx kydy
1
2
dy
dx ky
2/k x y
.2
dy y
dx x
2.
dy x
dx y
22 .
2
yx C
Example Ex. Suppose f is continuous and
Find f(x). Sol.
2
0( ) ( ) ln 2.
2
x tf x f dt
0( ) 2 ( ) ln 2
xf x f u du
( ) 2 ( ), (0) ln 2f x f x f
2( ) ln 2.xf x e
Homework 21 Section 8.2: 8, 14, 29
Section 8.3: 28, 29
Page 583: 7, 8, 10
Section 9.1: 10, 11