Differential Equations (Mathematical Modeling) - Revision history

Khanh at 21:33, 28 October 2021

2021-10-28T21:33:39Z

Lila: /* Resources */

2021-10-15T17:59:24Z

Resources

Lila: /* Exponential Growth and Decay */

2021-10-15T17:54:14Z

Exponential Growth and Decay

Lila at 17:53, 15 October 2021

2021-10-15T17:53:12Z

Lila at 17:50, 15 October 2021

2021-10-15T17:50:21Z

Lila at 17:47, 15 October 2021

2021-10-15T17:47:09Z

Lila: /* Resources */

2021-10-15T16:39:06Z

Resources

Lila at 16:37, 15 October 2021

2021-10-15T16:37:02Z

Rylee.taylor: added powerpoint by professor Roberts

2020-08-18T19:38:09Z

added powerpoint by professor Roberts

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.

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 ==Resources==
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.
-* [https://en.wikibooks.org/wiki/A-level_Mathematics/MEI/DE/Introduction_to_Differential_Equations/Using_Differential_Equations_in_Modelling Using Differential Equations in Modelling], Wikibooks: Introduction to Differential Equations (A-level Mathematics)
-* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Separable_2 Separable equations], Wikibooks: Ordinary Differential Equations
+==Licensing==
-* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Simple_Harmonic_Motion Simple Harmonic Motion], Wikibooks: Ordinary Differential Equations
+Content obtained and/or adapted from:

@@ Line 190: / Line 190: @@
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.
 * [https://en.wikibooks.org/wiki/A-level_Mathematics/MEI/DE/Introduction_to_Differential_Equations/Using_Differential_Equations_in_Modelling Using Differential Equations in Modelling], Wikibooks: Introduction to Differential Equations (A-level Mathematics)
-* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Simple_Harmonic_Motion Simple Harmonic Motion], Wikibooks: Differential Equations
+* [https://en.wikibooks.org/wiki/Ordinary_Differential_Equations/Separable_2 Separable equations], Wikibooks: Ordinary Differential Equations

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 where P is the population. That looks like the equation for exponential growth, doesn't it?  As a matter of fact, change the 'delta' to a differential, and it is.  The growth factor is (B-D).
-==== Example 3 ====
+'''Example'''
 :In a certain population of rabbits, the birth rate is 10%.  The death rate is 15%.  The initial population is 100.  How many rabbits are there after 10 years?  Will we always have rabbits?

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 Or, finally rearranging the result, substituting <math>\omega = \sqrt{k/m}</math>, and solving for <math>x</math> we obtain
 <p align="center"><math>x = \sqrt{\frac{2E}{k}}\cos (\omega t + \varphi)</math></p>
 == Other Applications==

← Older revision		Revision as of 17:50, 15 October 2021
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	Or, finally rearranging the result, substituting <math>\omega = \sqrt{k/m}</math>, and solving for <math>x</math> we obtain		Or, finally rearranging the result, substituting <math>\omega = \sqrt{k/m}</math>, and solving for <math>x</math> we obtain
	<p align="center"><math>x = \sqrt{\frac{2E}{k}}\cos (\omega t + \varphi)</math></p>		<p align="center"><math>x = \sqrt{\frac{2E}{k}}\cos (\omega t + \varphi)</math></p>
		+
		+	== Other Applications==
		+
		+	Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider the propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.

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 In order to solve any problem successfully, you need to develop a mathematical model that describes the situation adequately. In this module you will learn how to solve problems which can be solved using differential equations.
-With some problems you can go straight into the mathematics. This is usually the case where you have already have some models that you can apply to the situation. These usually include; but are not limited to newtons laws of motion, and many kinematics problems (stuff in mechanics 1, basically.)Because of this your solution to a problem will depend on assumptions and simplifications you made; like ignoring air resistance, or assuming constant mass, ect. This can sometimes give significant errors in your solution. Colecting data can help see if your model is off or not.
+With some problems you can go straight into the mathematics. This is usually the case where you have already have some models that you can apply to the situation. These usually include; but are not limited to newtons laws of motion, and many kinematics problems (stuff in mechanics 1, basically.)Because of this your solution to a problem will depend on assumptions and simplifications you made; like ignoring air resistance, or assuming constant mass, ect. This can sometimes give significant errors in your solution. Collecting data can help see if your model is off or not.
-Some problems will need their own model devolped, however. This means you will need to do some experiments to find a corrolation between the results you get. This will help with understanding the problem better, and help you formulate a mathematical model.
+Some problems will need their own model developed, however. This means you will need to do some experiments to find a correlation between the results you get. This will help with understanding the problem better, and help you formulate a mathematical model.
-===Example===
+==Examples==
-You have been asked to find out how much of an antidepressant is left in the body after a given time of taking the pill. How would you do this? Well the process of removing a drug from the body varies with the drug. The kidney plays the most important rule, doing something called renal clearance.The rate of this can be measured by taking urine samples.
+===Pharmaceutical example===
 Lets make some assumptions about the problem:
@@ Line 50: / Line 51: @@
 we replace the proportional sign:
-<math>\frac{dq}{dt} = -kq </math> where k is a postive constant of proportionality, and the negitive sign is there because the ammount of drug is decreasing.
+<math>\frac{dq}{dt} = -kq </math> where k is a positive constant of proportionality, and the negative sign is there because the amount of drug is decreasing.
 We will show later that the solution to the differential equation is:
@@ Line 86: / Line 87: @@
 <math> q=40e^{-0.0513k} </math>
@@ Line 91: / Line 138: @@
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.
 * [https://en.wikibooks.org/wiki/A-level_Mathematics/MEI/DE/Introduction_to_Differential_Equations/Using_Differential_Equations_in_Modelling Using Differential Equations in Modelling], Wikibooks: Introduction to Differential Equations (A-level Mathematics)

← Older revision		Revision as of 16:39, 15 October 2021
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	==Resources==		==Resources==
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.
		+	* [https://en.wikibooks.org/wiki/A-level_Mathematics/MEI/DE/Introduction_to_Differential_Equations/Using_Differential_Equations_in_Modelling Using Differential Equations in Modelling], Wikibooks: Introduction to Differential Equations (A-level Mathematics)

← Older revision		Revision as of 16:37, 15 October 2021
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		+	==Introduction to Differential Equations==
		+
		+	===Key points===
		+	*A differential equation is an equation involving a derivative.
		+	*The order of a differential equation is the order of the highest derivative.
		+	*Differential equations are used to model situations which involve rates of change.
		+	*The solution to a differential equation gives a relationship between the variables themselves, not the derivatives.
		+	*The general solution of a first order differential equation satisfies the differential equations and has a constant of integration in its solution.
		+	*The particular solution of a differential equation is one in which additional information has been used to calculate the constant of integration.
		+	*The general solution may be represented by a family of curves and the particular solution is one of that family.
		+	*To verify that a function is a particular solution, you must check that it satisfies the differential equation and initial conditions.
		+	How does the quantity of drug in the body vary with time? How long does it take a cup of coffee take to cool? After how many days will the moon take the same shape again? These question all have on thing in common. They have a quantity that is continuously changing. We see change in most things in life. Some changes are permanent; like the temperature of coffee. Some have repeating patterns, like the cycle of the moon.
		+
		+	You have already studied a lot of mathematics to describe these changes, but never really put that into real life applications. You will know from previously studies calculus that the rate of change of a quantity is called a derivative. If an equation contains a derivative, it is a differential equation.
		+
		+	The simplest form of a differential equation is to show the rate of change of one variable with respect to another like:
		+
		+	*1.<math>\frac{dy}{dx} = x + y</math>,
		+	*2.<math>\frac{dP}{dt} = 0.02P</math>
		+
		+	These are 2 very basic examples of differential equations. You will learn how to solve them in later sections.
		+
		+	A solution to a differential equation gives a relationship between the variables, which doesn't involve a derivative. The solutions to the differential equations above are given by:
		+
		+	*1.<math>y=-x-1+Ce^x</math>
		+	*2.<math>P=Ae^{0.02t}</math>
		+	(where C and A are constants of integration)
		+
		+	Differential equations begin to get more complicated where higher derivatives are involved (<math>\frac{d^2y}{dx^2}</math>), or if they included more variables.
		+
		+	==Modeling==
		+	In order to solve any problem successfully, you need to develop a mathematical model that describes the situation adequately. In this module you will learn how to solve problems which can be solved using differential equations.
		+
		+	With some problems you can go straight into the mathematics. This is usually the case where you have already have some models that you can apply to the situation. These usually include; but are not limited to newtons laws of motion, and many kinematics problems (stuff in mechanics 1, basically.)Because of this your solution to a problem will depend on assumptions and simplifications you made; like ignoring air resistance, or assuming constant mass, ect. This can sometimes give significant errors in your solution. Colecting data can help see if your model is off or not.
		+
		+	Some problems will need their own model devolped, however. This means you will need to do some experiments to find a corrolation between the results you get. This will help with understanding the problem better, and help you formulate a mathematical model.
		+
		+	===Example===
		+	You have been asked to find out how much of an antidepressant is left in the body after a given time of taking the pill. How would you do this? Well the process of removing a drug from the body varies with the drug. The kidney plays the most important rule, doing something called renal clearance.The rate of this can be measured by taking urine samples.
		+
		+	Lets make some assumptions about the problem:
		+	*Once the pill has been taking, it is instantly absorbed into the body
		+	*The drug is removed from the bloodstream via renal clearance.
		+	*The rate of renal clearance is proportional to the quantity of the drug in the body
		+
		+	We can now formulate a model. Let 't' be time in hours, and 'q' be the amount of drug in the body (in mg). this means that:
		+
		+	<math>\frac{dq}{dt} \propto q </math>
		+
		+	we replace the proportional sign:
		+
		+	<math>\frac{dq}{dt} = -kq </math> where k is a postive constant of proportionality, and the negitive sign is there because the ammount of drug is decreasing.
		+
		+	We will show later that the solution to the differential equation is:
		+
		+	<math>q=Ae^{-kt}</math>, Where A is a constant of integration. Although we have not shown this solution is true, we can check that is is a solution by differentiating it.
		+
		+	<math>\frac{dq}{dt} = A(-ke^{-kt}) = -kAe^{-kt} </math>
		+
		+	<math> q=Ae^{-kt} </math>
		+
		+	<math> \frac{dq}{dt}=-kq </math>, so therefore we have shown that it is a solution
		+
		+	Now we need to find values for A and k,for this we will need some experimental results:
		+
		+	*In the morning a man takes 40 mg of the antidepressant
		+	*Every hour, approximately 5% of it is gone
		+
		+	We now let t=0, and q=40
		+
		+	<math> 40=Ae^{0}=A </math>
		+
		+	in this case our constant of integration, A, is 40. So therefore:
		+
		+	<math> q=40e^{-kt} </math>
		+
		+	Now lets find k. We know that every hour the amount of drug decreases by 5% every hour, so when t=1, q=(0.75)*40=38
		+
		+	<math> 38=40e^{-k} </math>
		+
		+	Dividing each side by 40, and taking logs of each side gives that:
		+
		+	<math> k=-ln(\frac{38}{40})= 0.0513 </math>
		+
		+	This gives that:
		+
		+	<math> q=40e^{-0.0513k} </math>
		+
		+
		+	==Resources==
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.		* [https://mathresearch.utsa.edu/wikiFiles/MAT1193/Differential%20Equations%20(Mathematical%20Modeling)/Presentation17_Setting%20Up%20Differential%20Equations.pptx Setting up Differential Equations]. PowerPoint file created by Professor Cynthia Roberts, UTSA.