Transcript 3 For the differential equation xy

Problem of the Day
What are all values of x for which the function f
2
-x
defined by f(x) = (x - 3)e is increasing?
A) There are no such values of
x
B) x < -1 and x > 3
C) -3 < x < 1
D) -1 < x < 3
E) All values of x
Problem of the Day
What are all values of x for which the function f
2
-x
defined by f(x) = (x - 3)e is increasing?
A) There are no such values of
x
B) x < -1 and x > 3
C) -3 < x < 1
D) -1 < x < 3
E) All values of x
Differential Equations
Differential equations are very common in
physics, engineering, and all fields involving
quantitative study of change. They are used
whenever a rate of change is know but the
process giving rise to it is not.
Because the derivative is a rate of change, such an equation
states how a function changes but does not specify the
function itself. Given sufficient initial conditions, however,
such as a specific function value, the function can be found
by various methods, most based on integration.
Suppose that you are a bungee jumper, standing on a
bridge somewhere in Colorado. Three hundred feet
below you, a lovely little stream meanders through a
scenic canyon. The lovely stream is six inches deep,
and has lots of sharp pointy rocks in it. You have
brought a variety of cords with which to secure your
feet, ranging in stiffness from a steel cable to a soft
rubber band. Every one of the cords is exactly 160
feet long, when hung from the bridge. If you choose a
cord that is too stiff, then your body will no longer
form a connected set after you hit the end of the
cord. On the other hand, if you choose one that is too
soft, your body may still be moving for a short time
after you pass the point 300 feet below where you are
standing. Which cord should you choose, if any?
This physical system was studied by famous bungee
jumper Robert Hooke about 300 years ago. Hooke's
Law states that the force exerted on your body by the
cord is proportional to the distance of your body past
the equilibrium position of the spring.
We formulate a two-equation system of differential equations to model beaver
migration according to the recently formulated ``social-fence'' hypothesis of
small mammal dispersion. This hypothesis can be viewed as the ecological
analog of osmosis: Beavers from an environmentally superior habitat are posited
to diffuse through a social fence to an inferior but less-densely populated habitat
until the pressure to depart (``within-group aggression'') is equalized with the
pressure exerted against invasion (``between-group aggression''). This is termed
``forward migration.'' Assuming that the controlled parcel is a superior habitat,
the owner must be concerned with the ``backward migration'' that occurs when
the superior parcel becomes less densely populated through trapping.
Irrigation and Conservation
A single linear differential equation can be used to
investigate the question of when conservation of water by
agriculture is useful and genuine.
General and Particular Solutions
A function y = f(x) is called a solution of a
differential equation if the equation is
satisfied when y and its derivatives are
replaced by f(x) and its derivatives.
y' + 2y = 0
differential equation
y = Ce
general
solution
-2x
y = 5e
-2x
particular solution
The order of a differential equation
is determined by the highest-order
derivative in the equation.
y' + 2y = 0
y'' = -32
First-order
Second-order
Verifying Solutions
Determine whether the function is a
solution of the differential equation y'' - y =
0.
y = sin x
y = 4e
-x
Verifying Solutions
Determine whether the function is a
solution of the differential equation y'' - y =
0.
y = sin x
y' = cos x
y'' = -sin x
y'' - y = -sinx - sinx = -2sinx =
0
it is not a
solution
Verifying Solutions
Determine whether the function is a
solution of the differential equation y'' - y =
0.
-x
-x
y = 4e
y' = -4e
-x
-x
-x
y'' = 4e
y' - y = 4e - 4e =
0
it is a solution
The general solution of a first-order
differential equation represents a family of
curves known as solution curves, one for
each value assigned to the arbitrary value.
C = -2
C
y= x
C=2
C = -1
C=1
C = -1
C=2
C=1
C = -2
Particular solutions of a differential equation
are obtained from initial conditions that give
the value of the dependent variable for a
particular value of the independent variable.
s''(t) = -32
differential
equation
2
s(t) = -16t + C
t+C
2
1
general
solution
s(0) = 80
s'(0) =
initial conditions
64
2
s(t) = -16t + 64t + 80
particular solution
For the differential equation xy' - 3y = 0, verify
that
y = Cx is a solution and find the particular
solution with initial condition y = 2 when x = -3
For the differential equation xy' - 3y = 0, verify
3
that y = Cx is a solution
and find the particular
solution with initial condition y = 2 when x = -3
3
y = Cx
2
2
y' = 3Cx
3
x(3Cx ) - 3(Cx ) = 0
0 = 0 It is a
solution
For the differential equation xy' - 3y = 0, verify
3
that y = Cx is a solution
and find the particular
solution with initial condition y = 2 when x = -3
3
y = Cx
3
2 = C(-3)
-2 = C
27
3
Particular solution is y = 2x
27
Solving a differential equation analytically can
be difficult or even impossible. There is a
graphical approach that you can use called
sketching a slope field which shows the general
shape of all the solutions.
At each point (x,y) in the xyplane y' = 1/x determines the
slope of the solution at that
point.
x
y
y'
1
2
3
4
5
6
1
1
1
1
1
1
1
1/2
1/3
1/4
1/5
1/6
Sketch the slope field for the following
differential equation
y' = xy
x
y
y'
Sketch the slope field for the following
differential equation
y' = xy
2
Sketch the solution for the equation y' = 1/3 x 1/2 x
that passes through (1, 1)
Sketch the solution for the equation y' = y +
xy
that passes through (0, 4)
Sketch the solution that passes through
the point (0, 1/2)
y' = xy
Sketch the solution that passes through
the point (0, 1/2)
y' = xy
Attachments