First Order Linear DEs, Applications

Transcription

1 Week #2 : First Order Linear DEs, Applications Goals: Classify first-order differential equations. Solve first-order linear differential equations. Use first-order linear DEs as models in application problems. 1

2 First-order Linear Equations - 1 First-order Linear Equations How does one integrate first-order linear equations? A linear first-order equation is one that can be expressed in the form a 1 (x)y + a 0 (x)y = b(x) Assuming that a 1 (x) 0, we can rewrite the equation in the standard form: dy + P (x)y = Q(x). dx Note: when P (x) and Q(x) are constants, this equation is also separable (seen earlier).

3 First-order Linear Equations - 2 Problem. Solve dy dt + 2y = 3.

4 First-order Linear Equations - 3 Although separation of variables method cannot be used in general, Euler popularized an alternative: Multiply the standard form by an appropriate integrating factor so that resulting equation is easily integrable. More specifically, we multiply our standard form by by M(t) to obtain M(t) dy + 2M(t)y = 3M(t) dt Can we choose M(t) so that the left side is simply the derivative of one function?

5 Integrating Factors - 1 Integrating Factors For dy + P (t) y = Q(t), the integrating factor is defined to be dt ( ) M(t) = e P (t) dt. Show how multiplying the DE by M(t) results in the left side being the derivative of a product.

6 Integrating Factors - 2 Problem. Solve dy dx + 2y = 3ex. Note: this is a linear DE, but it is not separable.

7 Integrating Factors - 3 dy dx + 2y = 3ex.

8 Integrating Factors - 4 Problem. Verify that the solution found satisfies dy dx + 2y = 3ex.

9 Problem. Solve the initial value problem where Linear DEs - IVP Example dy xdx 2y x 2 = x cos(x) and y ( π ) 2 = 3

10 Linear DEs - IVP Example dy xdx 2y x 2 = x cos(x) and y ( π ) 2 = 3

11 Linear DEs - IVP Example 1-3 Verify that the solution found satisfies 1 dy xdx 2y x 2 = x cos(x).

12 Problem. Solve tx + 2x = 4t 2 and x(1) = 2. Linear DEs - IVP Example 2-1

13 Linear DEs - IVP Example 2-2 tx + 2x = 4t 2 and x(1) = 2.

14 Linear DEs - IVP Example 2-3 Problem. Verify that the solution found satisfies tx +2x = 4t 2 and x(1) = 2. tx + 2x = 4t 2 x(1) = 2

15 Linear DEs - Examining Solution Properties - 1 Problem. Let a and λ be positive constants and let b be any real number. Show that every solution to y +ay = be λt has the property that y 0 as t.

16 Linear DEs - Examining Solution Properties - 2

17 Linear DEs - Examining Solution Properties - 3

18 Applications - Tank - 1 Applications In Week 1, we noted that many physical, chemical and biological behaviours can naturally be represented by differential equations. In the next few examples, we study the types of predictions that can be inferred from solutions to DEs that arise in applications.

19 Applications - Tank - 2 Problem. At time t = 0 a tank contains Q 0 kg of salt dissolved in 100 litres of water. Assume that water containing 1/4 kg of salt per litre is entering the tank at a rate of r litres per minute, and that the well-stirred mixture is draining from the tank at the same rate. (1) Set up the initial value problem that describes this flow process.

20 Applications - Tank - 3 (2) Find the amount Q(t) of salt in the tank at any time. Find the limiting amount Q L.

21 Applications - Tank - 4

22 Applications - Tank - 5 (3) If r = 3 and Q 0 = 2Q L, then find the time T after which the salt level is within 2% of Q L. Find the flow rate that is required if the value T is not to exceed 45 minutes.

23 Applications - Tank - 6

24 Applications - Gravity-Fed Water Flow - 1 Problem. A titration buret (a tall, narrow cylinder used to measure liquids) has a hole in the bottom, out of which the liquids flow. If the buret is filled with water, the height of the water in the buret drops at a rate proportional to the square root of the current water height. (1) Let h be the height of water in buret. Write the differential equation for the rate of change of h with respect to the current height. (2) The differential equation in (a) has a constant of proportionality in it. Is that constant positive or negative?

25 Applications - Gravity-Fed Water Flow - 2 (3) At what height h does the water level become constant? Explain why this height is reasonable in the context of the problem.

26 (4) Find the general solution to the differential equation. Applications - Gravity-Fed Water Flow - 3

27 Applications - Gravity-Fed Water Flow - 4 (5) The buret originally is filled up with water to a height of 16 cm, and after 10 seconds the water level has dropped to a height of 4 cm. Determine the time at which the buret will be empty.

28 Applications - Pond - 1 Problem. Consider a pond that initially contains 10 million litres of fresh water. Water containing an undesirable chemical flows into the pond at the rate of 5 million litres per year; the mixture in the pond flows out at the same rate. The concentration γ(t) of chemical in the incoming water varies periodically with time according to the expression γ(t) = 2 + sin(2t) g L 1. (1) Construct a mathematical model of this flow process.

29 Applications - Pond - 2 (2) Determine the amount of chemical in the pond at any time.

30 Applications - Pond - 3

31 Applications - Pond - 4 (3) Describe the effect of the variation in the incoming concentration.

32 Applications - Pond - 5

33 Applications - Pond - 6

34 Applications - Pond - 7 Remark. Our model rests on several implicit assumptions. First, the amount of water is controlled entirely by the rates of flow in and out none is lost by evaporation or by seepage into the ground or gained by rainfall. Second, the same is also true of the chemical none is absorbed by fish or other organisms. Third, the concentration of chemical is uniform throughout the pond. Whether the results obtained from the model are accurate depend strongly on the validity of these assumptions.