Math 266: Ordinary Differential Equations

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1 Math 266: Ordinary Differential Equations Long Jin Purdue University, Spring 2018

2 Basic information Lectures: MWF 8:30-9:20(111)/9:30-10:20(121), UNIV 103 Instructor: Long Jin Office hour: Tu 9:00-11:00, W 10:30-11:30 or by appointment, MATH 448. Course webpage: Section webpage: Textbook: Elementary Differential Equations and Boundary Value Problems (10th edition), by W. Boyce and R. Diprima

3 Introduction In this course, we study basic ordinary differential equations.

4 Introduction In this course, we study basic ordinary differential equations. Differential equations Differential equations are mathematical equations that relate unknown functions with their derivatives. Differential equations often arise as mathematical model for physical systems.

5 Introduction In this course, we study basic ordinary differential equations. Differential equations Differential equations are mathematical equations that relate unknown functions with their derivatives. Differential equations often arise as mathematical model for physical systems. Classification Ordinary differential equations vs. Partial differential equations. Single differential equations vs. System of differential equations. Order of differential equations. Linear differential equations vs. Nonlinear differential equations.

6 Mass-spring system The motion of a mass attached to a spring and a damper is described by the equation m d 2 x 2 + B dx + kx = F (t). This is a second order linear ordinary differential equation.

7 Pendulum The motion of a simple pendulum is described by the equation ml d 2 θ + mg sin θ = 0. 2 This is a second order nonlinear ordinary differential equation.

8 Vibrating string The motion of a vibrating string is described by the equation µ 2 u t 2 T 2 u x 2 = 0. This is a second order linear partial differential equation.

9 Maxwell equations The basic equations for electric and magnetic fields are Maxwell s equations This is a first order linear system of partial differential equations.

10 Navier Stokes equations In fluid mechanics, the Navier Stokes equations are used to describe the motion of viscous fluids. v t + (v )v = p + ν( v) + f(x, t), v = 0. This is a second order nonlinear system of partial differential equations.

Volterra equations dx dy = ax αxy, = cy + γxy.

11 Predator-Prey model The dynamics of biological systems in which a predator and a prey interact is described by Lotka Volterra equations dx dy = ax αxy, = cy + γxy. This is a system of first order nonlinear ordinary differential equations.

12 Lorenz system Lorenz system is a simplified model for atmosphere convection dx = σ(y x), dy = x(ρ z) y, dz = xy βz. This is a system of first order nonlinear ordinary differential equations.

13 Chua s circuit dx = α(y x f (x)), dy = x y + z, dz = βy. This is a system of first order nonlinear ordinary differential equations.

14 First order differential equations A general first order equation is of the form F (t, y, dy ) = 0 or in explicit form dy = f (t, y). Here y = y(t) is the unknown function and t is the variable. F (or f ) are some functions of three(or two) variables.

15 First order differential equations A general first order equation is of the form F (t, y, dy ) = 0 or in explicit form dy = f (t, y). Here y = y(t) is the unknown function and t is the variable. F (or f ) are some functions of three(or two) variables. Different approaches Algebraic: Find exact solution, not always possible. Geometric: Direction field, phase plane, etc. Analytic: Analyzing solutions without solving. Numeric: Approximate values for solutions.

16 Example: A falling object As a first example, we consider the motion of a falling object.

17 Example: A falling object As a first example, we consider the motion of a falling object.

18 Example: A falling object As a first example, we consider the motion of a falling object. x(t): the distance it travels after time t v(t) = x (t): the velocity at time t a(t) = v (t): the acceleration at time t m > 0: mass g = 9.8m/s 2 : gravity constant

19 Example: A falling object As a first example, we consider the motion of a falling object. x(t): the distance it travels after time t v(t) = x (t): the velocity at time t a(t) = v (t): the acceleration at time t m > 0: mass g = 9.8m/s 2 : gravity constant Then by Newton s second law m dv = ma = F g = mg or dv = g.

20 Example: A falling object As a first example, we consider the motion of a falling object. x(t): the distance it travels after time t v(t) = x (t): the velocity at time t a(t) = v (t): the acceleration at time t m > 0: mass g = 9.8m/s 2 : gravity constant Then by Newton s second law m dv = ma = F g = mg or dv = g. By direct integration, v(t) = v 0 + gt. x(t) = v 0 t gt2.

21 Example: A falling object in the air A more realistic model considers also the air resistance (drag force).

22 Example: A falling object in the air A more realistic model considers also the air resistance (drag force).

23 Example: A falling object in the air A more realistic model considers also the air resistance (drag force). The equation becomes m dv = F g F d = mg F d.

24 Example: A falling object in the air A more realistic model considers also the air resistance (drag force). The equation becomes m dv = F g F d = mg F d. The drag force F d depends on the velocity v: When v is small, Stokes s formula F d = γv. In general, Rayleigh s formula F d = 1 2 ρv 2 C d A. But C d may also depend on v and other parameters.

25 Example: A falling object in the air Assume that v is small, then the equation of motion becomes Say m dv = mg γv. m = 10kg, g = 9.8m/s 2, γ = 2kg/s, we have a first order ordinary differential equation dv = v.

26 Example: A falling object in the air Assume that v is small, then the equation of motion becomes Say m dv = mg γv. m = 10kg, g = 9.8m/s 2, γ = 2kg/s, we have a first order ordinary differential equation dv = v. We can no longer solve this by direct integration v = ( v). (Next time: solve this using a specific technique.)

27 Geometric point of view In general, for a first order differential equation dy = f (t, y), it may not even be possible to get solutions in explicit form.

28 Geometric point of view In general, for a first order differential equation dy = f (t, y), it may not even be possible to get solutions in explicit form. Geometrically The unknown function y = y(t) can be represented by its graph in ty-plane y = dy is the slope of the tangent line of the graph at (t, y(t)) Although we do not know the graph, the equation tell us the tangent line at each point (t, y) has slope f (t, y).

29 Direction field The direction field for the equation dy = f (t, y), is a field of small straight line segments such that There is a small line segment at each point (t, y) The slope of the line segment at (t, y) is f (t, y)

30 Direction field The direction field for the equation dy = f (t, y), is a field of small straight line segments such that There is a small line segment at each point (t, y) The slope of the line segment at (t, y) is f (t, y) Then the graph of a solution must be tangent to all the line segment it passes through. Of course, it is impossible to draw the whole direction field But we can draw as many as we can and try to connect the line segments to get the graph of the solutions.

31 Direction field The direction field for the equation dy = f (t, y), is a field of small straight line segments such that There is a small line segment at each point (t, y) The slope of the line segment at (t, y) is f (t, y) Then the graph of a solution must be tangent to all the line segment it passes through. Of course, it is impossible to draw the whole direction field But we can draw as many as we can and try to connect the line segments to get the graph of the solutions. The direction field can also help us analyze the behavior of the solution. It is also related to the basic numerical method.

32 Direction field: Example Back to the equation The direction field looks like dv = v

Direction field: Example Note that when v < 49, dv = 9.

33 Direction field: Example Note that when v < 49, dv = v > 0 when v = 49, dv = v = 0 when v > 49, dv = v < 0

34 Equilibrium solution In particular, there is a solution v(t) 49. This is a constant function, often called an equilibrium solution.

35 Other solutions For other solutions to dv = v If v < 49, then v > 0 and v is increasing If v > 49, then v < 0 and v is decreasing In fact, all the solutions are monotone and converge to 49 when t. We can use the software dfield to plot the direction field and the solutions:

36 A second example: Population of field mice Consider the population p = p(t) of the field mice in a rural area. A simple model assumes that the growth of the population is proportional to the population itself where r > 0 is the growth rate. dp = rp

37 A second example: Population of field mice Consider the population p = p(t) of the field mice in a rural area. A simple model assumes that the growth of the population is proportional to the population itself where r > 0 is the growth rate. dp = rp But we also want to consider the fact that the owls are killing the mice constantly with rate k > 0. The equation becomes dp = rp k.

38 A second example: Population of field mice Consider the population p = p(t) of the field mice in a rural area. A simple model assumes that the growth of the population is proportional to the population itself where r > 0 is the growth rate. dp = rp But we also want to consider the fact that the owls are killing the mice constantly with rate k > 0. The equation becomes dp = rp k. This turns out to be not very different from the equation for a falling object dv = g γ m v.

39 A simple general equation Both equations are of the following general form with some constants a and b. dy = ay b

40 A simple general equation Both equations are of the following general form with some constants a and b. Next times dy = ay b Solve such kinds of equations. General feature about the solutions of differential equations. Applications: a real-life situation of falling object.

Elementary Differential Equations

Elementary Differential Equations George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 310 George Voutsadakis (LSSU) Differential Equations January 2014 1 /