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by: Alice Hsu

MATH 23 WEEK 2 NOTES Math 23

Marketplace > Dartmouth College > Math > Math 23 > MATH 23 WEEK 2 NOTES
Alice Hsu

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week 2 covers modeling with first order diferential equations. we go over autonomous equations, logistic differential equations, exact equations, mixture and compound interest word problems.
Differential Equations
Vardayani Ratti
Class Notes
Math, Differential Equations, Exact Differential Equations, Exact DE, Autonomous ODE, Logistic Growth and Decay Models, Mixtures, compound interest
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This 3 page Class Notes was uploaded by Alice Hsu on Saturday September 24, 2016. The Class Notes belongs to Math 23 at Dartmouth College taught by Vardayani Ratti in Fall 2016. Since its upload, it has received 4 views. For similar materials see Differential Equations in Math at Dartmouth College.


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Date Created: 09/24/16
MATH 23 WEEK 2 NOTES MODELING WITH DIFF EQ Solvent-solute mixture problems: goal is to compute the amount (Q(t)) of solvent in the tank at the time t. ???????? ???? ) ???????? = ???????????????? ???????? − ???????????????? ???????????? Assumptions:  Volume of water (V(t)) is constant if rate in = rate out  Concentration of solvent is constant  Mixture is well stirred  Concentration of inflow = c lb/gal  Concentration of outflow is c(t) lb/gal ???? ????) ???? ???? = ???? ????) Units are important!!!!!!!!!!!!!! Finances: Some S amount of money is invested at rate r, annually. S(t) is invested at time t. S(t) is dependent on rate of interest and principle. ???????? ????) = ???????? ± ???? ???????? AUTONOMOUS EQUATIONS An autonomous equation is one where the independent variable is not explicitly stated ???????? = ????(????) ???????? ???????? = ???????? − ???? is a special case of the autonomous equation ???????? Autonomous equations are separable. ???????? 1) If ???? ???? = 0, then ???????? = 0, ???? ???? = ???? ???????? 2) The solution of???????? = 0 is continuous for all points on the equation An exponential growth equation is the simplest form: ???????? = ???????? ???????? ???????? ???????? ???????? ∝ ????. Solution is ???? ???? = ???????? . There is exponential growth if r > 0, decay if r < 0. If 0 0 = ???? ,???? ???? = ???? ????^(????????). 0 But a population is restricted by resources, resulting in a carrying capacity. Thus, ???????? ???? = ????????(1 − ) ???????? ???? Which means that if the population is below the carrying capacity (k), then the inner term is 1. If y << k, y/k << 1. If y > k, y/k < 0. If y = k, dy/dt = 0. This kind of equation is known as a logistic differential equation. These equations are irritating to solve, so we study phase lines to determine behavior. The solutions of the equation are critical points. When y is less than k, dy/dt increases, when y is greater than k, dy/dt decreases. If growth and decay move towards a value, that is stable. If growth and decay move away from a value, that is unstable. Some problems converge on one side, but diverge on the other – these are semi-stable. EXACT EQUATIONS M(x,y)dx + N(x,y)dy is exact in ????:???? < ???? < ????,???? < ???? < ???? if F(x,y) such that ???????? ???????? (????,???? = ???? ????,???? ???????????? (????,???? = ????(????,????) ???????? ???????? For all (x,y) and R. If M(x,y)dx + N(x,y)dy is an exact differential form then ( ) ( ) M x,y dx + N x,y dy = 0 Or ???????? ???? ????,???? ) = ???????? ???? ????,????) Is called an exact equation. We can test for exactness by testing the second derivatives. ???????? ????,???? ) ???????? ????,???? ) ???????? = ???????? ∀ ????,???? ) The solution is ???? ????,???? = ???? because: M x,y dx + N x,y dy = 0 ???????? ???????? ???????? (????,???? + ???????? ????,???? = 0 ???????? ????,???? = 0 ???? ????,???? = ???? But what if the solution is not an exact equation? We can make it into one by finding the correct integrating factor, which is either: ???????? − ???????? ???? ????,???? = ???????? ???????? ???? ????,????) Or ???????? − ???????? ???????? ???????? ???? ????,???? = ???? ????,???? ) Then we integrate as per usual: ???? ????,???? = ???? ∫ ???? ????,???? And multiply IF into the equation and solve as per usual.


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