MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §9.3 ODE Applications

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §9.2 First Order, Linear, Ordinary Differential Equations  Any QUESTIONS About HomeWork §9.2 → HW

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 3 Bruce Mayer, PE Chabot College Mathematics §9.3 Learning Goals  Use differential equations to model applications involving public health, orthogonal trajectories, and finance.  Explore the predator-prey model

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 4 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Consider a population of individuals amidst an outbreak of some disease, with fractions of the total population S susceptible, I immune, and D diseased.  One model for the spread of an epidemic is that the rate of change in the fraction of diseased individuals is jointly proportional to the number susceptible and diseased individuals.

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 5 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  (a) Supposing that the size of the population and value of I are fixed, write a differential equation modeling the change in fraction of diseased individuals over time.  (b) If a constant 10% of the total population is immune, initially 0.04% of the population has the disease, and after one week 0.1% of the population has the disease, find an equation giving the fraction of the population that is diseased after t days.

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  SOLUTION:  (a) As with many modeling problems, start by carefully translating the English Words into mathematics:  “the rate of change in the fraction of diseased individuals is jointly proportional to the fraction of susceptible and diseased individuals”

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Since EveryOne is Either Sick, Immune, or Diseases then S, I, and D are Percentages that must add up to 100%:  Then the fraction of susceptible individuals “leftover” after immune and diseased individuals are accounted for.  Thus the revised ODE

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 8 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  (b) With a constant 10% Immune, solve an initial value problem for the differential equation  Note that this eqn is Variable Separable

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Integrating the Separated eqn:  The complex AntiDerivative can be accomplished using the Table of Integrals #6 from Section 6.1:  Use Algebra on the above Equation to solve for D(t)

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Working to Isolate D(t)  Letting  Find  Remove ABS bars as expect D<0.9 (90%)

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Working to Isolate D(t)  Now Use Initial Condition:  In the D(t) eqn  Next, Solve the Above Eqn for Constant Exponential PreFactor, A

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Solving for A  Use A in D(t)  Now Use the 2 nd Temporal Condition

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  By 2 nd I.C.

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 14 Bruce Mayer, PE Chabot College Mathematics Example  Model Epidemic  Use the Values of A & k to construct the completed Function, D(t)

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 15 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  The price of gasoline and the number of purchased electric cars depend on one another. Assume that the rate of change in price of gasoline is a decreasing linear function of the price of electric cars. Similarly, the rate of change in the number of electric cars purchased is an increasing linear function of the price of gasoline.  (a) Model the relationships in rates of change as linked differential equations

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  (b) Say that gasoline increases by $1 per year in the absence of electric cars and the rate decreases by 5 cents for each additional thousand cars that are produced. Also, say that if gas were (magically) priced at $0/gal, there would be a growth rate of 3 thousand, with the rate increasing by 4 thousand for each $1 increase in the unit price of gas. Solve the differential equations implicitly.

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  SOLUTION:  (a) Again Very CareFully Translate the Word-Statement to Math Relations  “the rate of change in price of gasoline is a decreasing linear function of the price of electric cars” 

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  Now construct the differential equation for the change in the car price:  “the rate of change in the number of electric cars purchased is an increasing linear function of the price of gasoline” 

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 19 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  Thus have constructed TWO ODE’s in for G(t) and C(t) with 4 unknown constants: a, b, c, and f  More translation is in order to find values of the constants in the two ODEs:

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  “gasoline increases by $1 per year in the absence of electric cars and the rate decreases by 5 cents for each additional thousand NonGasoline cars produced”  Also  “if gas were priced at $0/gal, there would be a growth rate of 3 thousand, with the rate increasing by 4 thousand for each $1 increase in the unit price of gas”

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 21 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  Instead of Finding G(t) and C(t) determine an Implicit relation between the two dependent variables  Note that G depends on C, and C depends on G; i.e. the Equations are Coupled  Try one of

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 22 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  Find dG/dC:  Separating the Variables  Finding the AntiDerivatives

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 23 Bruce Mayer, PE Chabot College Mathematics Example  Linked ODE’s  The Final G & C Relation:  This relationship does not define a function, but it nevertheless is a predictable relation between gasoline price and electric car sales The graph in the CG plane is an ellipse

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 24 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.3 P16: Atmospheric Pressure P28: Predator-Prey

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 25 Bruce Mayer, PE Chabot College Mathematics All Done for Today Predator vs Prey Wolves vs. Elk

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 26 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 27 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 28 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 29 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 30 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 31 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-14_Sp14_sec_9-2_1st_Linear_ODEs.pptx 32 Bruce Mayer, PE Chabot College Mathematics