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Differential Equations

by: Donny Graham

Differential Equations MTH 235

Donny Graham
GPA 3.57


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This 3 page Class Notes was uploaded by Donny Graham on Saturday September 19, 2015. The Class Notes belongs to MTH 235 at Michigan State University taught by Staff in Fall. Since its upload, it has received 49 views. For similar materials see /class/207316/mth-235-michigan-state-university in Mathematics (M) at Michigan State University.

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Date Created: 09/19/15
January 27 2009 1 1 Exam 1 Review Here is a summary of the material required for Exam 1 Make sure that you can do all problems on the rst four HW sets and the problems in the book Subject Matter 1 First Order Linear Equations i i d a Equation Set y i dig 2 191 9 p pt7 9 W 109 are continuous b Solution Set it efpdt 1 y7ugdt0 u 2 Bernoulli Equations a Equation 2 191 gy n 7e 01 b Solution Set 1 yl get linear equation for 1 117npv 1 ing Solve for 1 with an arbitrary constant Set y 01 keeping the arbitrary constant 3 Separable Equations a Equation b Solution dy 7 W 7 WW tdt0 i tdt o January 27 2009 1 2 4 Homogeneous Equations a Equation dy M 9571 Mtg7 M7 N homogeneous of the same degree b Solution Set y xi get y m M gt Nam M17 m i N17 71 Separable Solve for 1 getting Fzv C So7 implicit solution for X7y F7 y C r 5 Exact Equations a Equation Mds Ndy 0 is exact if My N1 b Solution Write f fde 9y Then7 set N fy Mdy g y Solve for 9y and get ay Mdz 99 O 6 Integrating Factors a Equation Mds Ndy 07 My 7 N1 31 0 b Solution i My 7 N1 Mzi i 7 f t f l N M is unc1ono zaone The function u Mr is an integrating factor The equation MMW WNW 0 is exact Solve as above January 27 2009 1 3 Nm 7 M y is function of y alone M M The function u My is an integrating factor The equation MMW WNW 0 is exact Solve as above 7 Modeling a Exponential growth Continuously compounded interest i Equation Q rQ r gt 0 ii Solution Qt Q06 b Exponential decay Radioactive decay i Equation Q rQ r lt 0 ii Solution Qt Q06 Half life th 7 gt 0 c Mixing Problems Tank with uid and initial amount Q0 of chem ical External uid with certain concentration 01 weight in unit volume is added at rate n galsunit time or litersunit time Completely mixed uid has concentration 00 drained with rate To galsunit time or litersunit time i Equation Let Qt be amount of chemical in tank at time t Vt be the volume of mixed uid at time f If V5 is initial volume of uid in tank7 then Vt Vb n 7 r0t Note if n 71 then Vt V5 for all t


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