Begin Usage 1: Differential Equations Pure mathmatics

Hi, it's me again! I've used XHTML to create the table below! As I expand on my skills for writing code I may as well display my skills as a problem solver as well. We're going to get started on Differential Equations now! They, as we shall see, are very useful. So, read on.

My plan is to use the table constructed below in order to outline 25 problems/solutions pertaining to Differential Equations! So LET'S DO THIS!!!!

First as always let's pray: Angel of God my Guardian Dear, with whom God's love entrusts thee here. Ever this day be at my side to light and guard and rule and guide!! Guide me Jesus! Amen!

Option to go to the Electrical Engineering table...Electrical Engineering

Differential Equations Topics Table

Usage 1: Pure Mathematics	Usage 2: Business	Usage 3: Physics	Usage 4: Biology	Usage 5: Engineering
check for solutions ex1	Financial: intro to solutions ex1	Rocket Propulsion	Culture/Population Growth ex1	Electrical Circuits
check for solutions ex2	Business: Financial Trends ex2	Orthogonal Curves	Culture/Population Growth ex2	Capacitive Circuits Review
separable equations ex3	Business: Financial Trends ex3	Hanging Chain/Pursuit Curves	Culture/Population Growth ex3	Inductive Circuits Review
1st order linear equation ex4 & intro to 2nd order equations	Business:the Financial Market ex4	Radioactive Decay	Culture/Population Growth ex4	Inductive/Capacitive
Exact (and homogeneous) equations also integrating factors ex5, ex6, ex7	Business: Numerical methods (Euler, Runge-Kutta	Maxwell's Equations & Schrodinger Equation Undamped, Damped, and Forced Vibrations	Culture/Population Growth ex5	RLC Circuits and more differential equations

The Table above indicates math (Calculus) problems that I will be solving!

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Usage 1: Pure mathematics: Just what are differential Equations?!?

So, to get started you know that in normal math processes you solve an algebra equation to find numbers. Scientists use mathematics to model processes in Physics/Engineering matters. Why important? Well, you may have to model a process with where the variables of the process change, like filling a tank of water, or expelling fuel from a rocket as it moves. To do this, you will have to solve a differential equation. But, when solving a differential equation you are actually trying to find ... AN EQUATION AS THE SOLUTION rather than a number (or numbers)!! This makes differential equations both fun and challenging. You may check any Differential Equations textbook to check that what I'm saying is true. (However, Differential Equations Demystified (by Krantz) is a good text to look at as reference.

If you are new to Differential Equations you should get to know the equals sign, or =, very well. The reason, well in order to solve these equations our first solution method involves making one side of the equation equal the other side. The very appearance of a Differential Equation may be awkward for the newbie. What it involves is an equation defined in terms of a function and its derivatives. Often the entire thing is set to zero, such that terms are on one side of the equation and zero is on the other side. Thus, a strategy will involve using the original function, possibly a linear combination of functions, and finding their derivatives. Thus, it is left to the problem solver to ensure that the final expression indeed equals zero when the function and its derivatives are plugged in.

Example 1: Given the expressions for y1,y2, and y3 show that a linear combination of the terms satisfies the Differential equation given. I have written this solution out below.

Now having all three derivatives achieved it is time to PLUG INTO THE EQUATION!!!

Bringing it all home collect the coefficients in front of each of the exponents verifying that each is indeed zero. Since the product of zero and any other number is zero then the final expression is zero. Zero = Zero on both sides. Thus, the linear combination shown is indeed a solution to the differential equation.

Lucky for us the above example is indicative of the type of math needed to solve actual physics/engineering problems. As I work on this page it will improve over time. Next check out another example problem: Example 2. Go see it!!!

Now on to another example: Ex2...D.E. Ex2.

Or if you are really feeling lucky move onto the next problem. We shall see it will involve the SEPARABLE EQUATIONS!! Ok! Please Comment!!!

Separable Differential Equations example: Ex3...separable D.E. Ex3.

Now available: Linear Differential Equations!!!

Linear Differential Equations example: Ex4...Linear D.E. Ex4.

Now available: Exact Differential Equations!!!

Exact Equations: Ex5...Exact D.E. Ex5.

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Usage 2: Business: How do financial analysts make predictions?

The link to Business Applications... to Business Applic!!!.

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Usage 3: Physics: The actions of the Universe Explained!

The link to Rocket Propulsion...to Rocket Propulsion!!!.

The link to Orthogonal Trajectories...to Orthogonal Trajectories!!!

The link to the Hanging Chain...to Hanging Chain!!!

The link to Pursuit Curves!!!Pursuit Curves

The link to Maxwell's Equations...to Maxwell's Equations!!!

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Usage 4: Biology: Natural Phenomena explained by Mathematics!

The link to Biology Applications...to Biology Applic!!!.

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Usage 5: Engineering: Constructing the world around us with Mathematics!

The link to Engineering Applications...to Engineering Applic!!!.

The link to RLC circuits ... to RLC circuits!!!

The link to an R-L example ... to the R-L example!!!