9.1 Solving Differential Equations

Transcription

1 9.1 Solving Differential Equations What is a differential equation? Real-world examples: The order of a differential equation is the order of the that occurs in the equation. A differential equation is call linear if it can be written in the form a n (x)y (n) + a n 1 y (n 1) + + a 1 (x)y + a 0 (x)y = b(x). Examples: Classify each differential equation as linear or nonlinear and give the order. (a) y (x) = 3x 2 (b) (y ) 3 + y = sin x (c) d2 y dx 2 + y = ex (d) y y + y y = t (e) dy dt = t 2 (y 1) A function f is called a solution of a differential equation if the equation is satisfied when and its are substituted into the equation. If we require the solution to satisfies the condition y(t 0 ) = y 0, we call this condition an and the problem of finding a solution to this type of DE is called an. 1

2 EX: Verify that y = sin x cos x cos x is a solution of the initial value problem y + (tan x)y = cos 2 x y(0) = 1 on the interval π/2 < x < π/2. A separable equation is a first-order differential equation in which the expression for dy/dx can be factored as a function of x times a function of y, i.e dy dx = f(x)g(y). Example: Solve the differential equation: dy dx = 1 3y

3 Example: Solve the differential equation: dy dx = 4x 1 y. 12. Find the solution to the diff. eq. that satisfies the given initial condition. dp dt = P t,p (1) = 2 Example: Solve the IVP: dy dx = xy2, y(1) = 0. 3

4 9.3 Graphical Methods Many differential equations are impossible to solve in the sense of obtaining an explicit formula. Slope fields will give us a way to learn a lot about a solution via a graphical approach. Ex: Sketch a solpe field for the differential equation y = x 2 y 2. Ex: 8 p

5 5.8 Exponential Growth and Decay dy dt = ky So, in general differential equations of the form dy dt = ky with initial condition y(0) = y 0 have the solution: Population Growth Recall dp dt = kp. The relative growth rate of a population P is 1 dp P dt. EX: A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells. (a) Find the relative growth rate. (b) Find an expression for the number of cells after t hours. 5

6 Radioactive Decay Let m(t) be the mass remaining from and initial mass of m 0 of a radioactive substance after time t. It has been shown experimentally that dm dt = with k a negative constant. Thus, m(t) =. The half-life is the time required for half of a given quantity to decay. 8. Bismuth-210 has a half-life of 5.0 days. (a) A sample originally has a mass of 800 mg. Find a formula for the mass remaining after t days. (b) Find the mass remaining after 30 days. (c) When is the mass reduced to 1 mg? 6

7 Continuously Compounded Interest Formula for compounding n times a year: A(t) = A 0 (1 + r n )nt Formula for compounding continuously: 7

8 9.2 Models Involving y = k(y b) Given dy dt = k(y b), the general solution is. Newton s Law of Cooling Newton s Law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surrounding, provided that this difference is not too large. So, dt dt = EX: A fresh brewed cup of coffee has temperature 95 C in a 20 C room. When its temperature is 70 C, it is cooling at a rate of 1 C per minute. When does this occur? Falling Objects We can model objects free falling when air resistance is taken into account. v = k m (v + gm k ) Gives us a solution: And, taking the limit as t, we get: 8

9 EX: Let v(t) denote the velocity (m/s) of a 1 kg object dropped from the top of a tall building t seconds after its release. If air resistance at time t is 4v(t)kgm/s 2, (a) Find the formula for v(t). (b) What is the terminal velocity of the object? Annuities An annuity is an investment in which a principal P 0 is placed in an account earning interest at a rate r and money is withdrawn in regular intervals. EX: What is the minimum initial deposit necessary to fund an annuity for 20 years if withdraws are made at a rate of $20,000 dollars per year at an annual interest rate of 5%? 9