engr 213: applied ordinary differential...

ENGR 213: Applied Ordinary Differential Equations

Youmin ZhangDepartment of Mechanical and Industrial Engineering

Concordia UniversityPhone: x5741 Office Location: EV 4-109

Email: [email protected]://users.encs.concordia.ca/~ymzhang/courses/ENGR213.html

Chapter 2First-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)

ENGR213: Applied Ordinary Differential Equations

Lecture Notes on Lecture Notes on ENGR 213ENGR 213 Applied Ordinary Differential Equations, by Youmin Zhang (CU)Applied Ordinary Differential Equations, by Youmin Zhang (CU)Chapter 2Chapter 2 3

2.1 Solution Curves without a SolutionDirection Fields

Slope of the solution curve lineal elementDirection field

Autonomous First-Order DEsDEs free of the independent variableCritical points equilibrium pointsSolution curvesAttractors and repellers

Three ways of study of DEs: qualitatively, analytically, and numerically


2.1.1 Direction FieldsLineal Element

1)A solution y = y(x) of a 1st-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 1st 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 xy-plane 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.


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(b)

2.1.1 Direction Fields

Fig. 2.1(a)


Direction Field1) 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.



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

Example 1



Example 2: A Falling Hailstone (1/3) A hailstone has mass m = 0.025 kg and drag

coefficient = 0.007 kg/s. Taking g = 9.8 m/sec2, the differential equation for

the falling hailstone is

or

or

vmgdtdvm =0.025 (0.025)(9.8) 0.007dv v

dt=

9 .8 0 .28dv vd t

=

9.8 0.28v 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'0 9.85 8.410 715 5.620 4.225 2.830 1.435 040 -1.445 -2.850 -4.255 -5.660 -7

9.8 0.28v v =


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 0 :

9.8 0.28 09.80.2835

vv

v

v

= =

=

=

9.8 0.28v v =

Direction fields


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) Newtons law of cooling: Rate of change of u(t) is

proportional to the difference u(t) - T(t). Or

Note: k>0 If u>T, then du/dt must be negative

( )du k u Tdt

=


Let k=1.5, T(t)=60 + 15 sin(2t). Then

External temperature is solid curve

Note the lag time for the internal temperatureto respond to changes in external temperature

1.5( 60 15sin(2 ))du u tdt

=

Example 3: Heating and Cooling (2/2)

Direction fields


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 1st-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 1st-order equations

f (y) and f (x, y)

are autonomous and nonautonomous, respectively.

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.


Example 3: An Autonomous DE


1) The DE

where a and b are positive constants, has the normal form dP/dt = f(P), tx and Py in (1), hence is autonomous.

2) From f(P) = P(abP) = 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(abP) on these intervals and whether a nonconstant solution P(t) is increasing or decreasing on an interval.

)( bPaPdtdP

=


2.1.2 Autonomous First-Order DEs4) The following table explain the figure

Interval Sign of f(P) P(t) Arrow- Decreasing Down

+ Increasing Up

- Decreasing Down

Fig. 2.4 is called a one-dimensional phase portrait, of dP/dt = P(abP), or simply phase portrait. The vertical line is called phase line.

Fig. 2.4



Idea: Without solving an autonomous DE, we would like to say more on its solution curves.

1) If (x0, y0) is in a subregion Ri, 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)



Fig. 2.5

If y(x) is bounded above by a critical point (as in R1), 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 R2, 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 -.


2.1.2 Autonomous First-Order DEsExample 4: Example 3 Revisited

Fig. 2.6 Phase portrait and solution curves in each of three subregions


2.1.2 Autonomous First-Order DEsExample 5: Solution Curves of an Autonomous DE

Fig. 2.7 Behavior of solutions near y = 1 for an ODE2)1( = y

dxdy


2.1.2 Autonomous First-Order DEsAttractors and Repellers

a) Asymptotically stable: Fig. 2.8(a), or called as an attractorb) Unstable: Fig. 2.8(b) or referred toas a repellerc) Semi-stable: Fig. 2.8(c) & (d)

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:

Fig. 2.8 Critical point c is: an attractor (a), a repeller (b), and semi-stable (c) and (d)


Reading and ExerciseReading and Exercise Reading

Section 2.1 Assignment (Due May 17th to be handed in the tutorial session)

Section 2.1: 3,4,21,24,26,27 (3rd edition), orSection 2.1: 2,4,17,20,22,23 (2nd edition).

engr 213: applied ordinary differential...

Documents