Differential Equations and Modeling Motion

Transcript

1. MATH 251 §2.5: Differential Equations; Modeling Motion February 5, 2013
2. A differential equation is an equation involving derivatives of a function. Language for the laws of nature Describe a quantity by its rate of change Foremost subject in applied mathematics Large area of current research A solution of a differential equation is a function which satisfies the equation for all values of x in the domain.
3. Examples Some examples of differential equations: 1 For all x, we have y = 3x + 7. What is y? 2 For x > 0, we have that xy = 1. What is y? 3 For all x, we have that y = 1 y . What is y? 4 For all x between −1 and 1, y = − x y . What is y?
4. A physically relevant example An object is in freefall. If we measure distances in feet and time in seconds, we have the following functions: Quantity Symbols Unit Height y(t) feet Velocity y (t) feet per second Acceleration y (t) feet per second squared Galileo told us y (t) = −32.
5. Solving the differential equation If y (t) = −32, then y (t) = −32t + C1 , where C1 can be any number. Notice: y (0) = C1 . Similarly, we have y(t) = −16t2 + C1 t + C2 where y(0) = C2 . Remember the freefall function! h(t) = −16t2 + v0 t + h0
6. Initial Value Problem An initial value problem (IVP) is a description of a function by a differential equation and some data about the function at a particular point. Example dy dx = x2 y(3) = 4
7. Solving Initial Value Problems Solve the initial value problem dy dx = x2 y(3) = 4
8. Solving Initial Value Problems Solve the initial value problem dy dx = x2 y(3) = 4 If dy dx = x2, the power rule tells us that y = 1 3 x3 + C for some number C. What is C? y(3) = 4 1 3 33 + C = 4 9 + C = 4 C = −5.
9. Freefall IVP I An object is launched straight up at an initial velocity of 20 feet per second from the ground. Find a function y = h(t) which describes its height over time. When does the object hit the ground?
10. Freefall IVP I An object is launched straight up at an initial velocity of 20 feet per second from the ground. Find a function y = h(t) which describes its height over time. When does the object hit the ground?        y = −16 ft/s2 y(0) = 0 ft y (0) = 20 ft/s
11. Freefall IVP II An object was dropped from a high place. Some time later, it hits the ground at 45 feet per second. What is the function that describes its trajectory? From how high was it dropped? How long did it fall?        y = −16ft/s2 y(0) = 0 y (0) = −45
12. Population growth The United States grows at a rate of about 1% per year. As of July 2012, the population was 313 million. Write an initial value problem which describes the population. P = 0.01P P(2012.5) = 313, 000, 000
13. Newton’s Law of Cooling The rate at which an object cools is proportional to the difference of the termperature of the object and the temperature of its surroundings. T = −k(T − T0 ), k > 0 T(t): temperature of object T0 : temperature of surroundings (assumed constant) k: a constant depending on the material.
14. Boundary Value Problems Warning This material is not in the book. Sometimes a function is described by a differential equation and data at two different points; this is called a boundary value problem. Example        y = −16ft/sec2 y(0) = 0 y(10) = 0
15. Freefall BVP How fast must a ball be thrown upward to remain in the air for ten seconds? How high would it go?