Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 10.1: Mathematical Modeling: Setting up.

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Presentation transcript:

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 10.1: Mathematical Modeling: Setting up a Differential Equation Sections 10.1 Introduction to Differential Equations

Review: (Regular) Equations

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Definition: Differential Equation A differential equation is an equation in which a derivative of an unknown function is one of the terms.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Definition: Solution to a Differential Equation A solution to a differential equation is a function such that when it and/or its derivatives are substituted into the differential equation, the equation represents a true statement. #1: y= x 2 is not a solution #3: y=1 is another solution

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Definition: Initial Value Problem An initial value problem is a differential equation with an unknown function together with the value of that function at some point

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example Radioactive carbon (carbon - 14) decays at a rate proportional to the amount of carbon-14 present.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example A yam is placed inside a 200°F oven. The temperature of the yam increases at a rate proportional to the difference between the oven temperature and its temperature.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example Morphine is administered to a patient intravenously at a rate of 2.5 mg per hour. About 34.7% of the morphine is metabolized and leaves the body each hour.

Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example Josh's credit card debt grows at a rate of 13%. Right now he owes $1,347.17

Solve

It is less clear what to do symbolically. Try guessing constant solutions. Look at solutions graphically.

Direction Field - Slope Field

Qualitative Can any of the following curves represent a solution to this differential equation?

Qualitative Can the curve below be a solution to any of the following differential equations?

Finding Solutions