Calculus Animations,Graphics and Lecture Notes

Differential Equations

About Kelly Liakos
Calculus 1- Limits and Derivatives
Calculus 1 - The Second Derivative
Calculus 1 and 3 Formula Sheets
Chain Rule
Computer Lab Assignments
Conic Sections
Differential Equations
Directional Derivatives/Gradient
Double Integrals
Equillibrium Solutions 1st order DEs
1st order Diff Eqs -Motion
Flux Integrals and Surface Integrals
Infinite Sequences
Infinie Series
Level Curves and Level Surfaces
Line Integrals
Optimization and Related Rates
Optimization for Functions of 2 Variables
Parametric Equations 2-space
Parametric Equations 3-space
Partial Derivatives
Polar Coordinate System
Polar Coordinates- Derivatives and Integrals
Riemann Sums and the Fundamental Theorem of Calculus
2d order Diff EQS-Motion
2d Partial Derivatives
Supplemental Exercises and Solutions
Tangent Planes/ Differential for f(x,y)
Trigonometry Applications
Triple Integrals
Unit Tangent Vectors/Unit Normal Vectors
Vectors in General
Vector Valued Functions
Visualizing Limits of Functions of 2 Variables
Special Topics
Scratch Paper
2 Poems

The following Lecture Notes detail the use of Laplace Transforms in the solution of 1st and 2d order equations including an example with a discontinuous forcing function.

The following Notes show how to compute the Laplace Transform

of cos(at-b) 3 ways:

1. Using the basic transform formulas for cos(at) and sin(at)

2. Using the definition of Laplace Transform

3. Using Mathcad

Laplace Transform of cos(at-b)

Notes Using Laplace Transforms

The following are my lecture notes and animation on Newton's Law of Cooling

Notes - Newton's Law of Cooling

Animation Newton's Law of Cooling

The following 3 links are links to my pages on the application of Differential Equations to motion.

1st order Diff Eqs -Motion

2d order Diff EQS-Motion

Equillibrium Soultions to Autonomuos DEs

Notes - Vector Flow Fields

The following are my lecture notes and animations which deal with orthogonal trajectories.

Notes - Orthogonal Trajectories




Show that the tangent lines to the hyperbolas  x2 - y2 = k  and xy = p  are perpendicular at each point

of intersection for all non-zero values of p and q. (For a complete discussion of orthogonal trajectories

see my notes and animations on orthogonal trajectories on the differential equation page)

Notes - Orthogonal Hyperbolas

A very useful tool in Differential Equations is the Unit Step Function.

See the notes on this important tool in the following download.

The Unit Step Function Notes

The following four animations correspond to the functions discussed

in the unit step function notes.

Animation - Rectangular Pulses

Animation-Sawtooth Curve-3 Teeth

Animation-Sawtooth Curve-20 Teeth

Animation-Decaying Exponential

Animation-Sine Rectification

The following are the lecture notes and animation which deals with using Laplace Transforms and a series of retro rocket firings to land a probe  gently on the surface of Mars.

Notes - Landing on Mars

Animation-Landing on Mars

Suppose we have an inverted conical tank with height H and radius R


Suppose fluid is flowing through a hole in the bottom with cross sectional area a


 with velocity given by V(t) = k [2 g h(t)]1/2   where h(t) is the height of fluid in the tank.


 Find the time required to empty the tank

Solution to Conical Tank Problem

 A Mixture Application

A 50 litre tank is initially filled with 10 litres of brine solution containing 20 kg of salt. Starting from time t=0, distilled water is poured into the tank at a constant rate of 4 litres per minute. At the same time, the mixture leaves the tank at a constant rate of k^(1/2)  litre per minute,. The time taken for overflow to occur is 20 minutes. Find Q(t) the amount of salt in the tank 0 < t < 20

Solution to Mixture Problem

Using a Differential Equation to Prove an Identity

Using Power Series show the solution to


dy/dx + x = y    is   y = 1 + x + Cex .


  Verify this result using an appropriate integrating factor

Power Series Solution

Lecture Notes - Variation of Parameters

The Smart Bunny-A very short story by Kurt Vonnegut Jr.

©2008-2010 Kelly Liakos