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## MATH-S343/M343 Section 1.1 Notes

by: Kathryn Brinser

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# MATH-S343/M343 Section 1.1 Notes MATH-S343

Marketplace > Indiana University > Mathematics > MATH-S343 > MATH S343 M343 Section 1 1 Notes
Kathryn Brinser
IU
GPA 4.0

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Covers notes from the first week's lectures: exponential growth and decay with differential equations and an introduction to the concepts of them in general.
COURSE
Honors Differential Equations
PROF.
Michael Jolly
TYPE
Class Notes
PAGES
2
WORDS
CONCEPTS
math-s343, math-m343, Differential Equations, Exponential Functions
KARMA
Free

## Popular in Mathematics

This 2 page Class Notes was uploaded by Kathryn Brinser on Thursday August 25, 2016. The Class Notes belongs to MATH-S343 at Indiana University taught by Michael Jolly in Fall 2016. Since its upload, it has received 7 views. For similar materials see Honors Differential Equations in Mathematics at Indiana University.

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Date Created: 08/25/16
S343 Section 1.1 Notes- Types of Differential Equations; Exponential Growth/Decay 8-23-16  Exponential Growth and Decay o Rate of change- proportional to amount of “stuff” o Ex. Mouse population growth rate  ???? = ???????? ???????? ???? ???? = ???????????? ????????  ???? = 0.5 mice/month (decay rate)  Assume owls eat mice at rate of 15 mice/day  Let ???? ???? = number of mice at time ???? ????????  Rate of change of mouse population is ???????? = 0.5???? − 450  450 comes from 15 mice/day multiplied by assumed 30 days in a month (made units of rates match)  0 = 0.5 900 − 450 horizontal slope (asymptote) occurs when 900 mice present  0.5 500 − 450 = −200 very negative slope (population goes down) when less than 900 mice present to start with  0.5 1000 − 450 = 50 positive slope (population increases) when more than 900 mice present o Ex. Let ????(????) be the weight of a cell at time ????. Write a differential equation that gives a model to how the cell grows. Its rate of growth is proportional to its surface area, and assume it is spherical. 4  ???????? = 4???????? 2 ???? = ???????? 3 3  Relates weight to SA; assume density constant, so weight ????(????) should be proportional to volume  ???????? = ???????? 2/3 ???????? o Ex. Using ???? = ???????? for a falling 10-kg object with a drag of 2 kg/s, write a differential equation to represent the acceleration of the object.  Let down be positive direction  Let ???? = 2 ???????? = drag coefficient ???? ????  Let acceleration of gravity ???? = 9.82 ????  ???? = ???????? = ???????? − ????????  ????( ) = ???????? − ???????? ????????  (10 ???????? ( ???????? ????) = 10 ???????? (9.8 ) − (2 ????????)(???? ) all units become ???? = ????????∙???? ???????? ???? ????2 ???? ???? ????2  10 ???????? = 98 − 2???? ????????  ???????? = ???? = 9.8 − 0.2???? ????????  Types of Differential Equations o Ordinary differential equations (ODEs)- relates unknown function to its rate(s) of change (derivatives) with respect to 1 independent variable o Partial differential equations- more than 1 independent variable; unknown function in equation with partial derivatives ???????? ???? ????  Ex. Heat equation: = 2 ???????? ???????? ???????? − ????Δ???? + ???? ∙ ???? ???? + ∇???? = ????  Ex. Nauer Stokes fluid flow:{???????? where ???? = force 2 ????2∙ ???? =20  Let ???? = ???? ,???? ,???? ) Δ???? = ???? ???? + ???? ???? + ???? ???? 1 2 3 ????????2 ????????2 ????????2  Let ∇???? = ( ????????,???????? ,???? ) ???????? ???????? ???????? o Linear differential equations (LDEs)- ???? (????,???? ???? ,???? ???? ,…,???????? (???? ) = 0 is linear if ???? is a linear function of ???? ???? ,???? ???? ,…,???? (????)(????)  Can prove that ???? (????,???? 1 + ???? ???? 2???? ???? +1???? ???? ,…,????2( ) 1????)(???? + ???? 2???? (???? ) = ( ) ′( ) (????)( ) ( ) ′( ) (????)( ) ???? (????,????1???? ,???? ????1,…,???? 1 ???? ) + ???? (????,???? 2 ,???? ????2,…,???? 2 ???? )  If linear, equation can be put into form (????) (????−1 ) ′ ???????????? ????) (???? + ???? ????−1 (???? ???? (???? + ⋯+ ???? ???? 1 ???? + ???? ???? ???? ????0= ???? ????( ) ( )  Derivatives can only be multiplied by functions of independent variable (usually ????) ′ (????)  Cannot have ???? ???? ,???? ???? ,…,???? (???? inside any “fancy” functions (ie. exponentials/polynomials) or multiplied by each other  ????(????) may be complicated; can be any function ????????  Ex. = 0.5???? − 450 ???????? = 3 − 2???? ???????? = 9.8 − 0.2???? ???????? ????????  Ex. Cell growth model nonlinear (power of 2⁄3 ): = ???????? 2/3 ????????  Order- highest derivative that appears o Ex. ???? + 2???? + ???? = 7 second order LDE ′′′ ???? ′ 1 o Ex. sin???? ????) + ???? ???? + ( )???? = log???? third order LDE

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