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

IU

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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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