Science One March 6, 2017 Application of Integration: Differential - PowerPoint PPT Presentation

Science One March 6, 2017

Application of Integration: Differential Equations (last week physics lecture) "# 𝑀 "$ + 𝐽𝑆 = 0 "# + "$ = − , 𝐽 separable equation "# + # = − , 𝑒𝑢 + ln 𝐽 = − , 𝑢 + 𝐷 implicit expression, easy to invert 𝐽(𝑢) = ±𝑓 7 𝑓 8 + , $ I.C. 𝐽 0 = 𝐽 9 , so 𝐽(𝑢) = 𝐽 9 𝑓 8 : ; $

The logistic equation (see slides from Nov 1) "< "$ = 𝑠𝑄(1 − 𝑄) carrying capacity K=1, 𝑄 0 = 𝑄 9 . (use partial fractions) < < C Implicit solution: ln A8< = 𝑠𝑢 + ln | A8< C | < C Explicit solution: 𝑄 𝑢 = < C D(A8< C )E FGH

Examples of mathematical models We use physical laws to formulate a differential equation.

Free fall in a medium (Newton’s 2 nd law) 𝑛 𝑒𝑤 𝑒𝑢 = −𝑛𝑕 + 𝐸 Model 1: drag proportional to velocity, 𝐸 = −𝑙𝑤 (Nov 1 slides) Model 2: drag proportional to square of velocity, 𝐸 = 𝐿 𝑤 O (solvable using partial fractions)

Cooling problem Physical Law (Newton’s law of cooling): A hot object introduced into a cooler environment cools at a rate proportional to the temperature difference between the object and its surroundings (same law applies to warming). If a cup of coffee sitting in a room maintained at a temperature of 20C cools from 80C to 50C in 5 minutes, how much longer will it take to cool to 40C?

Cooling problem Physical Law (Newton’s law of cooling): A hot object introduced into a cooler environment cools at a rate proportional to the temperature difference between the object and its surroundings (same law applies to warming). If a cup of coffee sitting in a room maintained at a temperature of 20C cools from 80C to 50C in 5 minutes, how much longer will it take to cool to 40C? "P "P A. "$ = 𝑙𝑈 C. "$ = 𝑙(𝑈 − 20) "P "P B. "$ = 𝑈 − 20 D. "$ = 𝑙𝑈 − 20

Cooling problem: solution dT/dt = k(T-20) dT/(T-20) = kdt ln|T-20| = kt+C Find C from initial condition, T(0) = 80 à C = ln(60) Find k from other information, T(5) = 50 à k = -ln(2)/5 Solve for T: T(t) = 20 + 60 e -(ln2 t)/5 Find t* such that T(t*) = 40 à t* = 5ln3/ln2 min

Math on the crime scene Just before midday the body of an apparent homicide victim is found in a room that is kept at a constant temperature of 20C. At 12pm the temperature of the body is 35C and at 1pm it is 34.2C. Assume that the temperature of the body at the time of death was 36.6 C and that is has cooled in accord to Newton’s law of cooling. What was the time of death?

Crime scene: Solution dT/dt = k(T-20) dT/(T-20) = kdt ln|T-20| = kt+C Find C from initial condition, T(0) = 35 à C = ln(15) Find k from other information, T(1) = 34.2 à k = ln(14.2/15) Solve for T: T(t) = 20 + 15 e ln(14.2/15)t Find t*<0 such that T(t*)=36.6 à t* ≈ 2 hrs.

Population growth with immigration Consider a national population P ( t ) with constant birth and death rates a and b (in births or deaths per year per unit of population). Suppose there is a constant net immigration rate of I individuals entering the country annually. Find a differential equation satisfied by P ( t ). If P (0)= P 0 , find P ( t ).

Population growth with immigration Consider a national population P ( t ) with constant birth and death rates a and b (in births or deaths per year per unit of population). Suppose there is a constant net immigration rate of I individuals entering the country annually, find a differential equation satisfied by P ( t ). If P (0)= P 0 , find P ( t ). "< "$ = 𝑏 − 𝑐 𝑄 + 𝐽 A Implicit solution U8V ln 𝑏 − 𝑐 𝑄 + 𝐽 = 𝑢 + 𝐷 # # U8V 𝑓 U8V $ − Explicit solution 𝑄(𝑢) = 𝑄 9 + U8V

Diffusion of information Let N ( t ) denote the number of people (in a fixed population P ) who by time t have heard a certain news item spread by the mass media. Under certain common conditions, the time rate of increase of N will be proportional to the number of people who have not yet heard the news. Find a D.E. satisfied by N ( t ). Suppose N(0)=0, find an expression for N ( t ) in terms of k and P .

Diffusion of information Let N ( t ) denote the number of people (in a fixed population P ) who by time t have heard a certain news item spread by the mass media. Under certain common conditions, the time rate of increase of N will be proportional to the number of people who have not yet heard the news. Find a D.E. satisfied by N ( t ). "W A. "$ = 𝑙 𝑄 − 𝑂 "W B. "$ = 𝑙 𝑂 − 𝑄 "W C. "$ = 𝑙𝑂 "W "W D. "$ = 𝑙 P − N N [ans: "$ = 𝑙 𝑄 − 𝑂 ]

Drug concentration in the blood A drug is introduced into the bloodstream intravenously at a constant rate a and breaks down and is eliminated from the body at a rate proportional to its concentration in the blood. Find a D.E. that models the concentration of the drug in the blood at time t from the injection.

Drug concentration in the blood A drug is introduced into the bloodstream intravenously at a constant rate a and breaks down and is eliminated from the body at a rate proportional to its concentration in the blood. Find a D.E. that models the concentration of the drug in the blood at time t from the injection. "[ "\ = 𝑏 − 𝑙𝑦

A mixing problem Initially a tank contains 1,000 L of brine with 50 kg of dissolved salt. Brine containing 10 g of salt per litre is flowing into the tank at a constant rate of 10 L/min. If the contents of the tank are kept thoroughly mixed at all times, and if the solution also flows out at 10 L/min, how much salt remains in the tank at the end of 40 min?

A mixing problem Initially a tank contains 1,000 L of brine with 50 kg of dissolved salt. Brine containing 10 g of salt per litre is flowing into the tank at a constant rate of 10 L/min. If the contents of the tank are kept thoroughly mixed at all times, and if the solution also flows out at 10 L/min, how much salt remains in the tank at the end of 40 min? "[ [ "$ = 0.01 ^ 10 − 10 ^ A999 𝑦 𝑢 = 10 + 40𝑓 8$/A99 At 𝑢 = 40, 𝑦 40 = 10 + 40𝑓 8O/b ≈ 36.8 kg

Another mixing problem A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept well mixed and it is drained from the tank at a rate of 3 L/min. If y ( t ) is the amount of salt (in kg) after t minutes, find a D.E. satisfied by y ( t ). Is the D.E. separable?

Another mixing problem A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept well mixed and it is drained from the tank at a rate of 3 L/min. If y ( t ) is the amount of salt (in kg) after t minutes, find a D.E. satisfied by y ( t ). Is the D.E. separable? "[ [ "$ = 5 ^ 0.4 − 3 ^ A99DO$ This is not separable!

A rate of reaction problem In a chemical reaction that goes to completion in solution, one molecule of each of two reactants, A and B, combines to form a molecule of the product C. According to the law of mass action, the reaction proceeds at a rate proportional to the product of the concentrations of A and B in the solution. Find a D.E. that models the number x ( t ) of molecules per cm 3 of C present at time t .

A rate of reaction problem In a chemical reaction that goes to completion in solution, one molecule of each of two reactants, A and B, combines to form a molecule of the product C. According to the law of mass action, the reaction proceeds at a rate proportional to the product of the concentrations of A and B in the solution. Find a D.E. that models the number x ( t ) of molecules per cm 3 of C present at time t . 𝑒𝑦 𝑒𝑢 = 𝑙(𝑏 − 𝑦)(𝑐 − 𝑦)

A geometrical problem Find a family of curves each of which intersects every parabola with equation 𝑧 = 𝐷𝑦 O at right angles.

A geometrical problem Find a family of curves each of which intersects every parabola with equation 𝑧 = 𝐷𝑦 O at right angles. 𝑒𝑧 𝑒𝑦 = − 1 2𝐷𝑦 \ C is set by the equation of the family of parabolas: 𝐷 = [ i . "\ [ "[ = − O\ [ i 𝑧 O + O = 𝐿 this is a family of ellipses (orthogonal trajectories)