Definition: Suppose we have a first-order differential equation of the form dy dx = f (x, y). The function f is called the slope function or rate function. Definition: The value f (x, y) that the function f assigns to a point represents the slope of a line or a line segment called a lineal element. Definition: If we systematically evaluate f over a rectangular grid of points in the xy-plane and draw a line element at each point (x, y) of the grid with slope f (x, y), then the collection of all these line elements is called a direction field or a slope field of the differential equation dy/dx = f (x, y). 2 dy Exercise 1. Reproduce the given computer-generated direction field of dx = e−0.01xy . Then sketch, by hand, an approximate solution curve that passes through each of the indicated points: (a) y(−6) = 0 (b) y(0) = 1 (c) y(0) = −4 (d) y(8) = −4. (#2) Class Exercise 1. Reproduce the given computer-generated direction field of dy dx = (sin x) cos y. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points: (a) y(0) = 1 (b) y(1) = 0 (c) y(3) = 3 (d) y(0) = − 52 . (#4) Exercise 2. Use computer software (Desmos for example) to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. (a) y ′ = x + y (i) y(−2) = 2 (ii) y(1) = −3 dy (b) dx = y1 (i) y(0) = 1 (ii) y(−2) = −1. (#6,8) Class Exercise 2. Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. dy (a) dx = xey (i) y(0) = −2 (ii) y(1) = 2.5 dy (b) dx = 1 − xy (i) y(− 21 ) = 2 (ii) y( 32 ) = 0. (#10,12) Definition: An ordinary differential equation in which the independent variable does not appear explicitly is said to be autonomous. Example: An example of an autonomous differential equation is dy dx = y 3 + 2y + 1. Example An example of a nonautonomous differential equation is dy dx = x + y + 1. Definition: An autonomous differential equation can be written in normal form as dy dx = f (y). c is a critical point of an autonomous differential equation if it is a zero of f . Definition: A constant solution y(x) = c is called an equilibrium solution. Definition: A phase portrait is used to describe the nature of the equilibrium solutions for an autonomous differential equation. Definition: The vertical line in the phase portrait is called a phase line. Definition: Suppose c is a critical point for an autonomous differential equation. If the arrowheads on the phase line point toward c, c is asymptotically stable. Definition: Suppose c is a critical point for an autonomous differential equation. If the arrowheads on the phase line point away from c, c is asymptotically unstable. Definition: Suppose c is a critical point for an autonomous differential equation. If the arrowheads on the phase line point toward c on one side and away from c on the other side, c is asymptotically semi-stable. Exercise 3. Find the critical points and phase portrait of the autonomous first-order differential equations. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. (#22,24) dy (a) dx = y2 − y3 dy (b) dx = 10 + 3y − y 2 . Class Exercise 3. Find the critical points and phase portrait of the autonomous first-order differential equations. Classify each critical point as asymptotically stable, unstable, or semistable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. (#26,28) dy = y(2 − y)(4 − y) (a) dx y dy = ye e−9y . (b) dx y Translation Property: If y(x) is a solution of an autonomous differential equation dy/dx = f (y), then y1 (x) = y(x − k), k a constant, is also a solution. Homework: 1-13 (every 4th), 19, 21, 25, 29