Problem of the Day

What are all values of x for which the function f defined by f(x) = (x2 - 3)e-x is increasing?A) There are no such values of xB) x < -1 and x > 3C) -3 < x < 1D) -1 < x < 3E) All values of x

Problem of the Day

What are all values of x for which the function f defined by f(x) = (x2 - 3)e-x is increasing?A) There are no such values of xB) x < -1 and x > 3C) -3 < x < 1D) -1 < x < 3E) 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

y = 5e

-2x

differential equation

particular solution

general solution

y = Ce-2x

y' + 2y = 0

The order of a differential equation is determined by the highest-order derivative in the equation.

First-orderSecond-order

y'' = -32

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.

y = 4e

-x y' = -4e

-x-x y'' = 4e

y' - y = 4e - 4e = 0

-x-x

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.

y = Cx C = 1C = 2

C = -2C = -1C = 2

C = 1

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

s(t) = -16t + C t + C

22

1general solution

initial conditions

particular solution

differential equation

s(0) = 80 s'(0) = 64s(t) = -16t + 64t + 80

2

For the differential equation xy' - 3y = 0, verify thaty = 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 that y = Cx is a solution and find the particular solution with initial condition y = 2 when x = -3

y = Cx y' = 3Cx

3

3 2

x(3Cx ) - 3(Cx ) = 0

32

0 = 0 It is a 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

y = Cx

3

3

2 = C(-3)

3

-227= C

Particular solution is y = -2x

3

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 xy-plane y' = 1/x determines the slope of the solution at that point.

x y y'

1 1 12 1 1/23 1 1/34 1 1/45 1 1/56 1 1/6

y' = xy

Sketch the slope field for the following differential equation

x y y'

y' = xy

Sketch the slope field for the following differential equation

Sketch the solution for the equation y' = 1/3 x2 - 1/2 xthat passes through (1, 1)

Sketch the solution for the equation y' = y + xythat passes through (0, 4)

y' = xy

Sketch the solution that passes through the point (0, 1/2)

y' = xy

Sketch the solution that passes through the point (0, 1/2)

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