ENGR 213: Applied Ordinary Differential Equations

Transcription

1 ENGR 213: Applied Ordinary Differential Equations Youmin Zhang Department of Mechanical and Industrial Engineering Concordia University Phone: x5741 Office Location: EV

2 ENGR213: Applied Ordinary Differential Equations Chapter 2 First-Order Differential Equations Solution Curves without the solution (Section 2.1) Direction fields; Autonomous first-order DEs Separable Variables (Section 2.2) Linear Equations (Section 2.3) Exact Equations (Section 2.4)

3 2.1 Solution Curves without a Solution Direction Fields Slope of the solution curve lineal element Direction field Autonomous First-Order DEs DEs free of the independent variable Critical points equilibrium points Solution curves Attractors and repellers Three ways of study of DEs: qualitatively, analytically, and numerically 3

4 2.1.1 Direction Fields Lineal Element 1)A solution y = y(x) of a 1 st -order DE dy/dx = f (x, y)is necessarily a differential function on its interval I, it must also be continuous on I. Thus the corresponding solution curve on I have no breaks and must possess a tangent line at each point (x, y(x)). 2)The slope of the tangent line at (x, y(x)) on a solution curve is the value of the 1 st derivative dy/dx at this point, and this we know from the DE: f (x, y(x)). Suppose that (x, y) represent any point in a region of the xyplane over which the function f is defined. The value f(x, y) that the function f assigns to the point represents the slope of a line, a line segment called lineal element. 4

5 2.1.1 Direction Fields For example: consider the equation dy/dx=0.2xy, where f (x, y) = 0.2xy. At point (2, 3), the slope of a lineal element is f (2, 3) =1.2 (50.19 deg in angle). Fig. 2.1(a) shows a line segment with slope 1.2 passing through (2, 3). As shown in Fig. 2.1(b), if a solution curve also passes through the point (2, 3), it does so tangent to this line segment; in other words, the lineal element is a miniature tangent line at that point. Fig. 2.1(a) Fig. 2.1(b) 5

6 2.1.1 Direction Fields Direction Field 1) If we evaluate f over a rectangular grid of points in the xy-plane and draw a lineal element at point (x, y), then the collection of all these lineal elements is called a direction field or a slope field of the DE dy/dx = f (x, y). 2)Visually, the direction field suggests the appearance or shape of a family of solution curves of the DE, and consequently it may be possible to see at a glance certain qualitative aspect of the solutions. 3) A single solution curve that passes through a direction field must follow the flow pattern of the field; it is tangent to a line element when it intersects in the grid. 6

7 Example Direction Fields Fig. 2.2(a) DF for dy/dx = 0.2xy Fig. 2.2(b) Solution curves of 7

8 Example 2: A Falling Hailstone (1/3) A hailstone has mass m = kg and drag coefficient γ = kg/s. Taking g = 9.8 m/sec 2, the differential equation for the falling hailstone is or dv (0.025)(9.8) 0.007v dt = dv dt = v m dv dt = mg γ v or v = v 8

9 v = v Ex 2: Sketching Direction Field (2/3) Using differential equation and table, plot slopes (estimates) on axes below. The resulting graph is called a direction field. (Note that values of v do not depend on t.) v v'

10 Ex 2: Direction Field & Equilibrium Solution (3/3) When graphing direction fields, be sure to use an appropriate window, in order to display all equilibrium solutions and relevant solution behavior. Arrows give tangent lines to solution curves, and indicate where solution is increasing & decreasing (and by how much). Horizontal solution curves are called equilibrium solutions. Use the graph below to solve for equilibrium solution, and then determine analytically by setting v' = 0. Set v = 0 : v = 0 Direction fields v = v = v = v 10

11 Example 3: Heating and Cooling (1/2) A building is a partly insulated box The temperature fluctuations depend on the internal temperature, u(t), and the external temperature, T(t) Newton s law of cooling: Rate of change of u(t) is proportional to the difference u(t) - T(t). Or Note: k>0 du ku ( T) dt = If u>t, then du/dt must be negative 11

12 Example 3: Heating and Cooling (2/2) Let k=1.5, T(t)= sin(2πt). Then du dt = 1.5( u 60 15sin(2 πt)) External temperature is solid curve Note the lag time for the internal temperature to respond to changes in external temperature Direction fields 12

13 2.1.2 Autonomous First-Order DEs An ODE in which the independent variable does not appear explicitly is said to be autonomous. If x denotes the independent variable, then an autonomous 1 st -order DE can be written as or in normal form (1) Assume that f in (1) and its derivative are continuous functions of y on I. The following 1 st -order equations f (y) and f (x, y) are autonomous and nonautonomous, respectively. 13

14 2.1.2 Autonomous First-Order DEs Critical Points 1) If c is the zero of f in (1), i.e. f (c) = 0, we call c is a critical point of the autonomous DE (1), or equilibrium point, or stationary point. 2) If c is a critical point of (1), the y(x)=c is a constant solution of the autonomous DE. A constant solution y(x) = c of (1) is called an equilibrium solution. 3) Increasing or decreasing of a non-constant solution of y = y(x) can be determined by the algebraic sign of the derivative dy/dx. 14

15 2.1.2 Autonomous First-Order DEs Example 3: An Autonomous DE 1) The DE dp dt = P( a bp) where a and b are positive constants, has the normal form dp/dt = f(p), t x and P y in (1), hence is autonomous. 2) From f(p) = P(a-bP) = 0, we see that 0 and a/b are critical points and so the equilibrium solutions are P(t) = 0 and P(t) = a/b. 3) By putting the critical points on a vertical line, we divide the line into three intervals. The arrows on the line shown in Fig. 2.4 indicate the sign of f(p) = P(a-bP) on these intervals and whether a nonconstant solution P(t) is increasing or decreasing on an interval. 15

16 2.1.2 Autonomous First-Order DEs 4) The following table explain the figure Interval Sign of f(p) P(t) Arrow - Decreasing Down + Increasing Up - Decreasing Down Fig. 2.4 Fig. 2.4 is called a one-dimensional phase portrait, of dp/dt = P(a-bP), or simply phase portrait. The vertical line is called phase line. 16

17 2.1.2 Autonomous First-Order DEs Solution Curves Idea: Without solving an autonomous DE, we would like to say more on its solution curves. 1) If (x 0, y 0 ) is in a subregion R i, i=1, 2, 3, and y(x) is a solution whose graph passes through this point, then y(x) remains in the subregion for all x. 2) By continuity of f we must then have either f(y)>0 or f(y)<0 for all x in R i, i=1, 2, 3. In other words, f(y) cannot change signs in a subregion. 3) Since dy/dx = f(y(x)) is either positive or negative in a subregion, R i, i=1, 2, 3, a solution y(x) is strictly monotonic, i.e. cannot be oscillatory, nor can it have a relative extremum (maximum or minimum). 17

18 2.1.2 Autonomous First-Order DEs If y(x) is bounded above by a critical point (as in R 1 ), then the graph of y(x) must approach the graph of the equilibrium solution y(x)=c1 either as x or x -. If y(x) is bounded below by a critical point (as in R3), then the graph of y(x) must approach the graph of the equilibrium solution y(x)=c2 either as x or x -. If y(x) is bounded in R 2, then the graph of y(x) must approach the graphs of the equilibrium solutions y(x)=c1 and y(x)=c2 one as x and the other as x -. Fig

19 2.1.2 Autonomous First-Order DEs Example 4: Example 3 Revisited Fig. 2.6 Phase portrait and solution curves in each of three subregions 19

20 2.1.2 Autonomous First-Order DEs Example 5: Solution Curves of an Autonomous DE Fig. 2.7 Behavior of solutions near y = 1 for an ODE 2 = ( y 1) 20 dy dx

21 2.1.2 Autonomous First-Order DEs Attractors and Repellers Suppose y(x) is a nonconstant solution of the autonomous DE in (1) and c is a critical point of the DE. There are three types of behavior y(x) can exhibit near c: a) Asymptotically stable: Fig. 2.8(a), or called as an attractor b) Unstable: Fig. 2.8(b) or referred to as a repeller c) Semi-stable: Fig. 2.8(c) & (d) Fig. 2.8 Critical point c is: an attractor (a), a repeller (b), and semi-stable (c) and (d) 21

22 Reading Reading and Exercise Section 2.1 Assignment (Due May 17 th to be handed in the tutorial session) Section 2.1: 3,4,21,24,26,27 (3rd edition), or Section 2.1: 2,4,17,20,22,23 (2nd edition). 22