Class Notes 2: First Order Differential Equation Linear. MAE 82 Engineering Mathematics
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1 Class Notes 2: First Order Differential Equation Linear MAE 82 Engineering Mathematics
2 Introduction First Order Differential Equations First Order Differential Equation Solution Any differentiable function that satisfy this equation for all t in some interval is called a solution Objective Determine whether such function exist Develop Methods for funding them Linear equations (section 2.1) Separable equations (section 2.2) Exact equations (section 2.6)
3 Classification
4 General Form
5 Example 1 Integration
6 Example 1 Integration
7 Integration Factors Method dy dt ay g(t)
8 Integration Factors Method
9 Integration Factors Method
10 Integration Factors Method
11 Integration Factors Method
12 Integration Factors Method Example 3 dy dt 2y 4 t
13 Integration Factors Method Example 3
14 Integration Factors Method Example 3 y t Ce 2t y t t Review Examples 4 p.37, 5 p.38
15 Variation of Parameters Method dy dt p( t) y g( t)
16 Variation of Parameters Method
17 Variation of Parameters Method
18 Example - Variation of Parameters Method dy dt 2ty 2
19 Example - Variation of Parameters Method
20 Autonomous Equation Population Dynamics Passenger Pigeon The passenger pigeon or wild pigeon (Ectopistes migratorius) is an extinct North American bird. Named after the French word passager for "passing by", it was once the most abundant bird in North America, and possibly the world. It accounted for more than a quarter of all birds in North America. The species lived in enormous migratory flocks until the early 20th century, when hunting and habitat destruction led to its demise. Some estimate 3 to 5 billion passenger pigeons were in the United States when Europeans arrived in North America. Some reduction in numbers occurred from habitat loss when European settlement led to mass deforestation. Next, pigeon meat was commercialized as a cheap food for slaves and the poor in the 19th century, resulting in hunting on a massive and mechanized scale. A slow decline between about 1800 and 1870 was followed by a catastrophic decline between 1870 and Martha, thought to be the world's last passenger pigeon, died on September 1, 1914, at the Cincinnati Zoo.
21 Autonomous Equation Population Dynamics
22 Autonomous Equation Population Dynamics Introduction Autonomous Equations - Class of first order differential equations where the independent variable t does not appear explicitly. dy dt f (t) Applications Growth/Decline of population Medicine Ecology Global Economics Stability / Instability of the solution
23 Autonomous Equation Population Dynamics Introduction Types of differential equations Exponential Growth Logarithmic Growth Logarithmic Growth with Critical Threshold Logarithmic Growth with Threshold
24 Autonomous Equation Population Dynamics Exponential Growth The population of a given species at time t The rate of change of the population y is proportional to the current population (value of the y) Thaoms Maltus British Economist r The rate of growth (r>0) or decline (r<0) Solving the differential equation subject to initial condition
25 Autonomous Equation Population Dynamics Exponential Growth Under ideal condition the population will grow exponanatially Valid during a limited period of time Limitation Space Food supplies Limited resources
26 Autonomous Equation Population Dynamics Exponential Growth Human Population
27 Autonomous Equation Population Dynamics Exponential Growth Human Population
28 Autonomous Equation Population Dynamics Logistic Growth
29 Autonomous Equation Population Dynamics Logistic Growth
30 Autonomous Equation Population Dynamics Logistic Growth
31 Autonomous Equation Population Dynamics Logistic Growth
32 Autonomous Equation Population Dynamics Logistic Growth
33 Autonomous Equation Population Dynamics Logistic Growth f ( y ) r 1 y K y
34 Autonomous Equation Population Dynamics Logistic Growth dy y r 1 y f ( y) dt K
35 Autonomous Equation Population Dynamics Logistic Growth
36 Autonomous Equation Population Dynamics Logistic Growth
37 Autonomous Equation Population Dynamics Logistic Growth
38 Autonomous Equation Population Dynamics Logistic Growth
39 Autonomous Equation Population Dynamics Logistic Growth Human Population
40 Autonomous Equation Population Dynamics Logistic Growth Human Population
41 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold dy dt y r 1 y, r T 0
42 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold dy dt y r 1 y, r T 0 y y 0 y T T y 0 0 e rt
43 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold
44 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold Note 1: If the initial population y 0 is above the threshold T the graph of y(t) has a vertical asymptote at t* The population become unbounded in a finite time whose value depends on y 0, T and r
45 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold Note 2: The population of Species exhibit the threshold phenomena if (Below the threshold) Too few subjects are present, then the species can not propagate itself successfully and the population become extinct (Above the threshold) further growth occurs Note 3: The population cant become unbounded
46 Autonomous Equation Population Dynamics Logarithmic Growth with Threshold dy dt y y r 1 1 y, r 0 and 0 T K T K
47 Autonomous Equation Population Dynamics Logarithmic Growth with Threshold
48 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold Notes Passenger Pigeons were present in the US in vast numbers until the 19 th century It was heavily hunted for food and sport and reduced significantly 1880 Breed successfully only when present in a large concentration i.e. high number of T
49 Autonomous Equation Population Dynamics Logistic Growth with Critical Threshold Notes Large number of individual birds remained alive in the late 1880s. However there were not enough in any one place to permit successful breeding. The population decline to extinction Last survivor died in 1914 Solution Conservation
50 Linear Model Bacterial Growth
51 Linear Model Bacterial Growth
52 Linear Model Bacterial Growth Given: A culture initially has P 0 number of bacteria At t=1hr the number of bacteria is measured to be 3/2P 0 The rate of growth is propositional to the number of bacteria P(t) present at the time t Calculated The time necessary for the population to triple
53 Linear Model Bacterial Growth
54 Linear Model Bacterial Growth
55 Linear Model Bacterial Growth
56 Linear Model Bacterial Growth
57 Linear Model Bacterial Growth
58 Linear Model Cooling/Warming Newton s Law
59 Linear Model Cooling/Warming Newton s Law
60 Linear Model Cooling/Warming Newton s Law
61 Linear Model Cooling/Warming Newton s Law
62 Linear Model Cooling/Warming Newton s Law
63 Linear Model Series Circuits
64 Linear Model Series Circuits
65 Linear Model Series Circuits
66 Linear Model Series Circuits
67 Linear Model Series Circuits
68 Linear Model Series Circuits
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1+t 2 (l) y = 2xy 3 (m) x = 2tx + 1 (n) x = 2tx + t (o) y = 1 + y (p) y = ty (q) y =
DIFFERENTIAL EQUATIONS. Solved exercises.. Find the set of all solutions of the following first order differential equations: (a) x = t (b) y = xy (c) x = x (d) x = (e) x = t (f) x = x t (g) x = x log
Chapter 6 Reading Questions
Chapter 6 Reading Questions 1. Fill in 5 key events in the re-establishment of the New England forest in the Opening Story: 1. Farmers begin leaving 2. 3. 4. 5. 6. 7. Broadleaf forest reestablished 2.