Create Your Formula Sheet In Advance

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view some of the pictures linked to on this page.

QuickTime free download.

	What you know about math from YouTube
	911 math call from YouTube
	Here is a discussion of some of the terms used in differential equations from efunda, a site dedicated to engineering fundamentals.
	Maple Worksheet with differentiation and integration examples     audio/video
	Maple Worksheet on Partial Derivatives
	Maple Worksheet on Partial Derivatives and Multiple Integrals
	Maple Worksheet--Slope Fields
	Maple Worksheet for Examples 1 - 9
	Maple Worksheet for Examples 11 - 14 and 16
	PowerPoint Presentation of Solutions to Separable First Order Differential Equations
	PowerPoint Presentation of Solutions to Non-Separable First Order Differential Equations
	Springfield Technical Community College Differential Equations Distance Learning Project
	Here is a Terrific Site Frederick Bass found for me.  It is an interactive differential equations site from Addison Wesley Longman.  Check out what they have to offer on first order differential equations.
	Larry Green's terrific Lake Tahoe Community College differential equations notes and examples
	Here is a terrific set of notes for a differential equations course by Paul Dawkins at Lamar University.

Hopefully in Calculus I and Calculus II you learned to solve some ordinary differential equations (ODES) by the method of separation of variables.  To have a better sense of what the solution to a first order ODE is take a look at this First Order DE Solution Grapher (or this one or this one). You can "Google" Euler's Method and find lots of demos.  The picture on the right shows a blow-up of a portion of what the DE Solution Grapher will produce, in this case showing part of the graph of the solution to dy/dx = 2x, y(0) = 0.  The blue line segments indicate the direction of tangents to the graph of any solution to dy/dx = 2x that went through the left endpoint of the line segment. The Vanderbilt DE Toolkit (not working right now) can solve a variety of differential equations.

Try using the method of separation of variables to solve the following: dy/dx = y(cos(x)),      y(0) = 1           Solution The graph of the solution is shown on top at the right.  Click on the graph to see an animation of the direction field vectors moving across the screen for increasing values of x along with an animated solution point.  The lower graph on the right shows the solutions for y(0) = -1, -0.4, 0.4, and 1 along with the direction field.  Click on the graph to see an enlargement.  In addition to the graphs shown at the right you can look at various solutions corresponding to an initial condition of the form y(0) = c by following this link to a DPGraph of a surface and the plane y = c.  The curve of intersection of the surface and the plane when c = 1 is the graph of the solution to the initial value problem above.  You can use the scrollbar and activate c to look at solutions for various values of c.  You can also use the z-slice feature.

You need to be familiar with Theorem 1.2.1, page 15, in Zill dealing with existence and uniqueness of solutions to first order differential equations with a condition attached.

PowerPoint Presentation of Solutions to Separable First Order Differential Equations  

2.2  You will need to be able to solve a separable first order differential equation.  Here is another java applet DE solution grapher that also draws the direction field.

2.3  You will need to be able to solve a linear first order differential equation.

2.4  You will need to be able to solve a first order differential equation that is exact and also one that can be made exact by using a suitable integrating factor that is a function of one variable.

2.5  You will need to know some of the substitutions that can be made to transform a first order differential equation into a form in which it can be solved.  These would include the substitution that would turn a homogeneous equation (as defined on the top of page 71 in Zill) into one that is separable and the substitution that would transform a Bernoulli Equation (defined at the bottom of page 72 in Zill) into a linear equation.  See also how to turn an equation of the form of equation (5) on page 73 in Zill into one that is separable.

PowerPoint Presentation of Solutions to Non-Separable First Order Differential Equations

2.6  You will need to be able to understand and articulate in words the two simplest numerical methods for approximating discrete solutions to first order differential equations, Euler's Method and the Improved Euler Method.  Here is SOS Math on Numerical Methods.  I will demonstrate in class TI and Excel numerical solutions.  Here is a java applet demonstrating Euler's Method and a Second Order Runge-Kutta Method.  This link takes you to the UBC introduction to Euler's Method.  Here is a link to Euler's Method in Pictures.

Here are two more java applets demonstrating numerical methods of solving first order ordinary differential equations. This one uses Euler's Method and draws the direction fields in solving dy/dx = ax^m + byⁿ with 0 < x < 3 and 0 < y < 3.  This one demonstrates Euler, Improved Euler, and Runge-Kutta.

Another interesting approach (that you will not be tested on) is the Picard Iterative Process.

Click here to see some solutions to sample equations along with solution graphs.

3.1 and 3.2  You will need to be able to develop the differential equation to model exponential population growth and logistic population growth, radioactive decay, Newton's Law of Cooling, vertical motion with air resistance proportional to velocity (see projectile motion, the parachute example) and mixture problems.  This is another logistic growth link.  Here is an applet for discrete inhibited population growth and another one referenced as logistics population growth.  Here is a nice set of examples including applications involving separable differential equations found by Frederick Bass.  The examples include an excellent one on population growth.  Here are more examples including  Newton's Law of Cooling and mixture examples from the same source, Joseph Mahaffy from San Diego State University.  The link will take you to his Differential Equations site.  When you get there you will need to click on Lectures and then on Linear Differential Equations.  There are lots of other good DE things at this site too.

To the left is the graph of the solution to the logistics equation  dy/dx = .01y(100-y),   y(0) = 10. The solution is y = 100ex / (ex + 9).

PARACHUTE PROBLEM:  A man with a parachute jumps out of an airplane at an altitude of 5000 feet.  After 5 seconds his parachute opens and at the moment his parachute opens he catches a brief updraft.  The instant the first man's parachute opened a second man jumped out of the airplane at the same altitude.  The second man has the same weight and drag coefficient as the first man.  Regrettably the second man's parachute and back-up parachute both malfunctioned and did not open.     animation1--The first man with no scales     animation2--The first man with scales     animation3--Both jumpers, no scales, with the animation starting the moment the second man jumps     animation4--Both jumpers with scales Mathematics can be used to prove that there will be a limiting velocity, i.e., that there will be a limit to how fast you will fall.  There will be a limit to how fast you will fall even if you are not wearing a parachute but the limit will be a much larger number. To see a description of this complete with an animation go to free fall compared to using a parachute.  Another example can be found in the US Naval Academy parachuting applet.  Frederick Bass found these links:  Physics of Skydiving, Parachuting Animation, Another Parachuting Animation.

The graph above shows the height above the ground of the first man (parachute functions properly) as a function of time.  The picture on the right is by Derek Demeter.

Click here for a sample test problem on parachuting including the development of the velocity function and a few hints and helpful graphs and a link to the solution.

SUPER EC  Derive the position function for a projectile if, rather than neglecting air resistance, we represent air resistance (R) as proportional to velocity (v).  That is take R = -cv, some constant times velocity.  What this leads to, rather then starting out with r"(t) = <0,-g> where g is a positive constant (related to gravity) and y-positive is up, is a somewhat more complicated force equation.  For a freely falling body (or more safely perhaps a parachutist), if m stands for mass, a for acceleration, W for weight, then since net force F = ma equating forces yields ma = -W + R, taking the weight force to be in the negative direction and the air resistance force to be in the positive direction.  This could be written as ma = -mg - cv.  If k = c/m we get a = -g - kv.  For a position function r(t) = <x(t),y(t)> this translates into the following:

r"(t) = <-kx'(t),-g-ky'(t)>

Starting with r"(t) above and using the same initial conditions as used in deriving the position function shown in Theorem 12.3 of your calculus text (Larson, Hostetler, Edwards, Eighth Edition), derive the position function, r(t).  These initial conditions would be

SUPER SUPER EC

A winch at the top of a building is reeling in a rope that is attached to the end of a blue pipe.  The radius of the circular purple winch at the top of the building and the length of time it takes for it to make one complete revolution is given below. The height of the green building is 12 meters and the length of the blue pipe is 12 meters. In the animation the red circular path is that of the end of the pipe.  The blue point moving along the red circular path is indicating the speed the end point of the pipe would move at if it was moving at a constant speed and taking the required length of time to reach the top of the building.  The blue point on the small purple circle on top of the building is indicating the speed with which the winch is rotating.  Click here or on the picture at the right to see an animation.  Take the origin to be the point where the pipe touches the bottom of the building with the pipe initially lying along the positive x-axis.  Let (x,y) represent the coordinates of the endpoint of the pipe where the purple rope is attached and let z represent the length of the rope from the endpoint of the pipe to the corner of the building.  The extra credit is to find dx/dt and dy/dt when z = 6 meters and to find a parametric representation of the position function giving (x,y) as a function of t (in seconds).

Answers to problems 1-6 on an old Exam I

Powerpoint presentation:  Problems from an old Exam I  There is an error in an integral calculation in Number 5.  Click here for a correct solution.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats