Entrevista Ing. Rodrigo Regalado EVAP.

.[pic 1]Differential Equation Final Project

Due Date:

Team:  

Ricardo Arturo García Espinosa A01322960 IMD

Jahir Roberto Rivas Vázquez A01204988 IMT

Ramon Ariel Ivan Muñoz Corona A01330566 IMT

Grade:

Co Evaluation:

Average:

Professor: José Santiago González García

Campus Metropolitano de la Ciudad de México

Escuela de diseño, ingeniería y arquitectura

Introduction

NB. Sections denoted with [n] are cited directly from the corresponding reference

 Autonomous, first order differential equations

If in an ordinary differential equation doesn’t appear explicitly the independent variable, then it is called an autonomous differential equation. For example: are autonomous and non-autonomous respectively. [1][pic 2]

 Critical Points [1]

It is said that a real number c is the critical point of the autonomous differential equation is one of the roots of f, in other words, that . A critical point can also be called a stationary or equilibrium point. Saying this, if we put the constant function  in the equation, then both sides of the equation will become 0. This means that if c is a critical point of the equation, then  is a constant solution of the autonomous differential equation. [pic 3][pic 4][pic 5]

The constant solution  is known as the equilibrium solution; the equilibrium solutions are the only constant solutions of the equation. [pic 6]

Determining the algebraic sign of  we can determine when non-constant solution increases or decreases; on the  equation we can do it by identifying the intervals on y axis in with the function f(y) is positive or negative[pic 7][pic 8]

For example:

The differential equation:

[pic 9]

Where a & b are positive constants, has the normal form of . [pic 10]

We can observe that this equation has the same structure than  from the beginning. From  we can observe that 0 and  are critical points. Putting the points on a numeric line, we can see that we can find 3 intervals:[pic 11][pic 12][pic 13]

[pic 14][pic 15]

[pic 16]

[pic 17][pic 18][pic 19][pic 20][pic 21][pic 22][pic 23]

[pic 24]

The intervals are now:.[pic 25]

This table explains the figure:

Solution curves

“Without solving the differential equations, we can say a lot about its solution curve. Because the function f on the previous equation “dy/dx” Is independent from the variable “x”, we can say that f is defined for  or for . Also, since f and its derivative f’ are continuous functions of y on some interval I of the y-axis, the fundamental results of the unique solution existence theorem, hold in some horizontal strip or region R in the xy-plane corresponding to I, and so through any point (x0, y0) in R there passes only one solution curve of dy/dx. [pic 32][pic 33]

For making our analysis, let’s suppose that the equation has exactly 2 critical points C1 & C2 and that C1 is minor than C2. The graphs of y(x) =C1 and y(x) =C2 are horizontal lines; this 2 lines divide the region R in 3 regions “R1, R2, R3”

[pic 34][pic 35]

Some of the conclusions from drawing about a non-constant solution y(x) of dy/dx are:

• If (x0, y0) is in a sub region  and y(x) is a solution whose graph passes through this point, then y(x) remains in the sub region Ri for all x. The solution y(x) in R2 is bounded below by c1 and above by c2, that is, c1[pic 36]

• By continuity of f we must then have either f(y)>0 or f(y)< 0 for all x in a sub region Ri, i =1, 2, 3. In other words, f(y) cannot change signs in a sub region.

• Since dy/dx=f(y(x)) is either positive or negative in a sub region Ri, i=1,2, 3, a solution y(x) is strictly monotonic—that is, y(x) is either increasing or decreasing in the sub region Ri. Therefore y(x) cannot be oscillatory, nor can it have a relative extremum (maximum or minimum).

• If y(x) is bounded above by a critical point c1 (as in sub region R1 where y(x)< c1 for all x), then the graph of y(x) must approach the graph of the equilibrium solution y(x)= c1 either as  or as . If y(x) is bounded above and below by two consecutive critical points, then the graph of y(x) must approach the graphs of the equilibrium solutions y(x)=c1 and y(x) =c2, one as  and other as .  If y(x) is bounded below by a critical point, then the graph y(x) must approach the graph of equilibrium solution y(x)=c2 either as   or as . [pic 37][pic 38][pic 39][pic 40][pic 41][pic 42]

Example:

The autonomous equation  possesses the single critical point 1. From the phase portrait we conclude that a solution y(x) is an increasing function in the sub regions defined by and , where . For an initial condition , a solution y(x) is increasing and bounded by 1, and so ; for  as a solution y(x) is increasing and bounded. [pic 43][pic 44][pic 45][pic 46][pic 47][pic 48][pic 49]

Now y(x)=1-1/(x+c) is a one-parameter family of solutions of differential equations. A given initial condition determines a value for c. For the initial conditions, say, y(0)=-1<1 and y(0)=2>1, we find in turn, that y(x)=1-1/(x+1/2), and y(x)=1-1/(x-1). [pic 50]

The graph of each of these rational functions possesses a vertical asymptote. But bear in mind that the solutions of the IVPs

[pic 51]

Are defined on special intervals. They are respectively,

[pic 52]

The solution curves are portions of the graphs in b) and c) shown in blue. As predicted by the phase portrait, for the solution curve  as  ; for the solution curve in c),  as  from the left.” Cited directly from [2][pic 53][pic 54][pic 55][pic 56]

Attractors and repellers [1]

Suppose that y(x) is a non-constant solution of the autonomous differential equation dy/dx=f(x,y) and that C is a critical point of the differential equation. There are 3 types of behavior that y(x) can exhibit near our critical point:

1. When both arrowheads on either side of the dot labeled c point toward c, all solutions y(x) of dy/dx=f(x,y) that start from an initial point (x0,y0) sufficiently near c exhibit asymptotic behavior when . For this reason the critical point c is said to be asymptotically stable. In this case c is also referred as an attractor.[pic 58][pic 57]
2. When both arrow heads on either side of the dot c point away c, all solutions y(x) of dy/dx=f(x,y) that start from (x0,y0) move away from c as x increases. In this case the critical point c is said to be unstable. An unstable critical point is also called repeller.
3. &   d) The critical point illustrated on these cases neither an attractor nor a repeller. But since c exhibits characteristics of both, it is said that this point is semi-stable

Autonomous DEs and Direction Fields [1]

If a first-order differential equation is autonomous, then we see from the right-hand side of its normal form  that slopes of lineal elements that pass through points in the rectangular grid used to construct a direction field for the DE depend solely on the y-coordinate of the points. In other words, lineal elements passing through points on any horizontal line must all have the same slope; slopes of lineal elements along any vertical line will, of course, vary. For example:[pic 60][pic 59]

Stable Critical Points

Let X1 be a critical point of an autonomous system and X=X(t), the solution that satisfies X(0)=X0; where . It is said that X1 is a stable critical point when, for any given radius , there is a corresponding radius r>0 such that if the initial position X0 satisfies , the corresponding X(t) solution satisfies  for every t>0. If it also satisfies that  and that in all cases , it is said that X1 is an asymptotic sable critical point.[pic 67][pic 61][pic 62][pic 63][pic 64][pic 65][pic 66]

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