Section 2.1 : Solution Curves without a Solution Direction Fields (Slope Fields) - allow us to sketch solution curves to first-order ODEs. General first-order ODE: dy dx = f (x, y) (1) Let y = y(x) be a solution to this ODE on its interval I of definition. (Note that y is continuous on I since it is differentiable on I.) Let y(a)=b, where a I, so that (a, b) is a point on the solution curve. Then: f (a, b)= dy (a)= slope of the solution curve at (a, b) dx Therefore, the ODE (1) allows us to find the slope of a solution curve at every point (x, y) in a region of the xy-plane over which f is defined. The value of f (x, y) at point (x, y) represents the slope of the tangent line at (x, y). The small segment of the tangent line passing through (x, y) is called a lineal element. A direction field (or slope field) of ODE (1) is a graph of lineal elements on a set of points in the xy-plane (i.e., a graph of slopes of solution curves represented by line segments). 1
Example 1: Verify that the direction field below corresponds to the ODE y = 2 y. Then use this direction field to sketch approximate solution curves that pass through the points (a) y(0)=4, (b) y( 1)=0, and (c) y(3)=2. 6 4 2 0-2 -4-4 -2 0 2 4 6 2
Example 2: Use the direction field of the ODE y = x+ y to sketch approximate solution curves that pass through the points (a) y( 2)=2 and (b) y(1)= 3. 4 2 0-2 -4-4 -2 0 2 4 3
Example 3: Determine if the direction field corresponds to y = y, y = x/y, y = 1/y or y = 1/y 2. Then use the direction field to sketch approximate solution curves that pass through the points (a) y(0) = 1 and (b) y( 2) = 1. 4 2 0-2 -4-4 -2 0 2 4 4
Autonomous First-Order ODEs General first-order ODE: General first-order autonomous ODE: Examples: dy dx dy dx = f (x, y) (1) = f (y) (2) Step 1: Find critical points and equilibrium solutions: A real number c is called a critical point (or equilibrium point or stationary point) of an autonomous ODE (2) if f (c)=0. If c is a critical point of (2), then y(x)=cis a constant solution of (2). (Verify by substituting y(x) = c into (2).) A constant solution to (2) is called an equilibrium solution. Example: 5
Step 2: Determine phase portrait: Critical points are used to determine the (one-dimensional) phase portrait or phase line of an autonomous ODE. Example: 6
Step 3: Sketch solution curves: Phase portraits (or phase lines) can be used to sketch solution curves of autonomous ODEs. Since f=f (y) (i.e., f is independent of x) then f is defined for all meaningful x values, i.e., <x< or 0 x<. f and f f y are continuous on some interval I of the y-axis. the region R of Theorem 1.2.1 is some horizontal strip in the xy-plane corresponding to I, i.e. R={(x, y) : y I and <x< } or R={(x, y) : y I and 0 x< }. Therefore, through any point (x 0, y 0 ) in a horizontal strip R of the xy-plane, there is a unique solution to (2) in some interval containing x 0. 7
Step 3: Sketch solution curves (cont): Suppose y = f (y) has exactly two distinct critical points c 1 and c 2, where c 1 < c 2. The equilibrium solutions are y(x)=c 1 and y(x)=c 2 whose graphs are horizontal lines that partition R into three subregions R 1, R 2 and R 3 : R 1 ={(x, y) R : y<c 1 } R 2 ={(x, y) R : c 1 < y<c 2 } R 3 ={(x, y) R : y>c 2 } The following properties of nonconstant solutions y(x) of y = f (y) can be shown: 1. If (x 0, y 0 ) is in subregion R i (i=1, 2, 3) and y(x) passes through this point, then y(x) remains in subregion R i for all x. That is, the graph of a nonconstant solution cannot cross the graph of an equilibrium solution. 8
2. f (y) cannot change signs in a subregion R i (since it is continuous), i.e., f (y(x))>0for all x R i or f (y(x))<0for all x R i (i=1, 2, 3). Therefore, since dy/dx = f (y(x)), a nonconstant solution y(x) is strictly monotonic in subregion R i. That is, a nonconstant solution y(x) is either strictly increasing in R i or strictly decreasing in R i. Specifically, y(x) cannot oscillate or have local maxima or minima. 3. The graph of y(x) approaches the graph of one or more equilibrium solutions. (a) If y(x) is bounded above by critical point c 1 (i.e., y(x) is in R 1 ), then the graph of y(x) approaches the graph of the equilibrium solution y(x)=c 1 either as x or as x. (b) If y(x) is bounded below by critical point c 2 (i.e., y(x) is in R 3 ), then the graph of y(x) approaches the graph of the equilibrium solution y(x)=c 2 either as x or as x. (c) If y(x) is bounded above and below by critical point c 2 and c 1, respectively, (i.e., y(x) is in R 2 ), then the graph of y(x) approaches the graphs of the equilibrium solutions y(x)=c 2 and y(x)=c 1, one as x and the other as x. 9
Example: 10
Classifications of Critical Points: A critical point c is called: asymptotically stable if all solutions y(x) of (2) that start from an initial point (x 0, y 0 ) sufficiently close to y=c approach y=c as x, i.e., if lim y(x)=c x for any solution y(x) that starts sufficiently close to y=c. In this case, c is called an attractor. unstable (or a repeller) if all solutions y(x) of (2) that start from an initial point (x 0, y 0 ) sufficiently close to y=cmove away from y=c as x increases. semi-stable if some solutions of (2) that start sufficiently close to y=capproach y=cand some move away from y=cas x increases. Examples: 11
Example: 12