Mathematics 22: Lecture 1 - Introduction - …math.furman.edu/~dcs/courses/math22/lectures/lecture-1.pdfCourse information I Course information is available at: dcs/dokuwiki/ Dan Sloughter

Mathematics 22: Lecture 1Introduction

Dan Sloughter

Furman University

January 3, 2008

Dan Sloughter (Furman University) Mathematics 22: Lecture 1 January 3, 2008 1 / 16

Course information

I Course information is available at:http://math.furman.edu/~dcs/dokuwiki/


http://math.furman.edu/~dcs/dokuwiki/

Octave and Maxima

I Octave is a free software package for working with matrices.

I Windows version: http://prdownloads.sourceforge.net/octave/octave-2.1.73-1-inst.exe?download

I Other versions: see http://www.octave.org/download.html

I wxMaxima is a free computer algebra system.

I Windows version: http://sourceforge.net/project/downloading.php?group_id=4933&use_mirror=superb-west&filename=maxima-5.14.0a.exe&79647928

I Homework: install Octave and wxMaxima

I If you cannot install Octave and/or wxMaxima, see me about usingan account in the Mathematics Department computer lab.


http://prdownloads.sourceforge.net/octave/octave-2.1.73-1-inst.exe?download


http://www.octave.org/download.html

http://sourceforge.net/project/downloading.php?group_id=4933&use_mirror=superb-west&filename=maxima-5.14.0a.exe&79647928



Octave and Maxima















Octave and Maxima















Octave and Maxima















Octave and Maxima















Octave and Maxima















Octave and Maxima















Example in OctaveI The following creates a plot of y = t2 sin(t) on the interval [−5, 5]:

I t = [-5:0.01:5];I y = t.^2 .* sin(t);I plot(t,y)

I Note: ‘grid on’ will plot a gridI Use ‘replot’ to recreate the same graph (with new options)I Printing a graph:

I printI In Windows: Right click top window border, select print from options

menuI Saving a graph:

I gset terminal png;I gset output "filename.png";I replot;

I To reset the terminal in Windows for plotting to the screen:I gset terminal win

I To reset the terminal in the lab for plotting to the screen:I gset terminal x11




I Note: ‘grid on’ will plot a grid

I Use ‘replot’ to recreate the same graph (with new options)I Printing a graph:









I Note: ‘grid on’ will plot a gridI Use ‘replot’ to recreate the same graph (with new options)

I Printing a graph:I printI In Windows: Right click top window border, select print from options










menu

I Saving a graph:I gset terminal png;I gset output "filename.png";I replot;






















Example in Octave (cont’d)

I Alternative in newer versions:I print("temp.png","-dpng")

I See http://www.gnu.org/software/octave/doc/interpreter/Printing-Plots.html#Printing-Plots for other options.

I Printing in the lab:I gset terminal postscript;I gset output "|lpr";I replot;

I Exit octave with ‘quit’ or ‘exit’

I Plotting in wxMaxima: use menu item.

I Homework: Create, and print, some plots from Octave and fromwxMaxima.


http://www.gnu.org/software/octave/doc/interpreter/Printing-Plots.html#Printing-Plots



I Alternative in newer versions:I print("temp.png","-dpng")I See http://www.gnu.org/software/octave/doc/interpreter/

Printing-Plots.html#Printing-Plots for other options.






































Some terminology

I Technical definition: We call an equation of the form

du

dt= f (t, u),

where f (t, u) and u(t) may be vectors in Rn, a differential equation.

I In practice: a differential equation is a relationship between areal-valued function u of a single variable (usually t, for time) and itsderivatives.

I The order of a differential equation is the order of the highestderivative in the equation.


Some terminology


du

dt= f (t, u),





Some terminology


du

dt= f (t, u),





Examples

Idu

dt= t sin(u) is a first-order differential equation.

Id2u

dt2+ t2 du

dt+ cos(u) = cos(3t) is a second-order differential equation.


Examples

Idu

dt= t sin(u) is a first-order differential equation.

Id2u

dt2+ t2 du

dt+ cos(u) = cos(3t) is a second-order differential equation.


Solutions

I A solution of a differential equation is a continuous function which,with its derivatives, satisfies the relationship specified in the equation.

I Example:

I u = 4 cos(2t)− 3 sin(2t) is a solution to the equation

d2u

dt2= −4u.

I Verification:

d2u

dt2= −16 cos(2t) + 12 sin(2t) = −4(4 cos(2t)− 3 sin(2t)) = −4u.

I Note: this solution is not unique.I For example, both u = −8 sin(2t) and u = 23 cos(2t) are solutions as

well.


Solutions


I Example:

I u = 4 cos(2t)− 3 sin(2t) is a solution to the equation

d2u

dt2= −4u.

I Verification:

d2u



well.


Solutions


I Example:I u = 4 cos(2t)− 3 sin(2t) is a solution to the equation

d2u

dt2= −4u.

I Verification:

d2u



well.


Solutions



d2u

dt2= −4u.

I Verification:

d2u



well.


Solutions



d2u

dt2= −4u.

I Verification:

d2u


I Note: this solution is not unique.

I For example, both u = −8 sin(2t) and u = 23 cos(2t) are solutions aswell.


Solutions



d2u

dt2= −4u.

I Verification:

d2u



well.


Some terminology

I We say an equation in the formdu

dt= f (t, u) is in normal form.

I The equation is

I autonomous if f depends on u only.I linear if f is a linear function of u.

I Examples

Idu

dt= sin(u) is autonomous.

Idu

dt= t sin(u) is not autonomous.

Idu

dt= t2u + et is linear.

Idu

dt= −u2 is not linear (but is autonomous).


Some terminology



I The equation is

I autonomous if f depends on u only.I linear if f is a linear function of u.

I Examples

Idu


Idu


Idu


Idu



Some terminology



I The equation isI autonomous if f depends on u only.

I linear if f is a linear function of u.

I Examples

Idu


Idu


Idu


Idu



Some terminology



I The equation isI autonomous if f depends on u only.I linear if f is a linear function of u.

I Examples

Idu


Idu


Idu


Idu



Some terminology




I Examples

Idu


Idu


Idu


Idu



Some terminology




I Examples

Idu


Idu


Idu


Idu



Some terminology




I Examples

Idu


Idu


Idu


Idu



Some terminology




I Examples

Idu


Idu


Idu


Idu



Some terminology




I Examples

Idu


Idu


Idu


Idu



General and particular solutions

I Example: u = cet is a solution ofdu

dt= u for any value of c .

I Moreover, if u is a solution ofdu

dt= u, then u = cet for some value of

c .

I We call u = cet a one-parameter family of solutions to the equationdu

dt= u.

I Moreover, this family of solutions is the general solution ofdu

dt= u.

I Note: If we further require that u(0) = 10, then we find thatu(t) = 10et is the unique solution to the initial value problem

du

dt= u,

u(0) = 10.

I We call u(t) = 10et a particular solution ofdu

dt= u.







c .


dt= u.


dt= u.


du

dt= u,

u(0) = 10.


dt= u.







c .


dt= u.


dt= u.


du

dt= u,

u(0) = 10.


dt= u.







c .


dt= u.


dt= u.


du

dt= u,

u(0) = 10.


dt= u.







c .


dt= u.


dt= u.


du

dt= u,

u(0) = 10.


dt= u.







c .


dt= u.


dt= u.


du

dt= u,

u(0) = 10.


dt= u.


Solutions

I Example: The initial value problemdu

dt=√

u,

u(0) = −10

has no solutions.


dt= u

23 ,

u(0) = 0

has an infinite number of solutions, including u ≡ 0 and u = 127 t3.


Solutions


dt=√

u,

u(0) = −10

has no solutions.


dt= u

23 ,

u(0) = 0

has an infinite number of solutions, including u ≡ 0 and u = 127 t3.


Partial derivatives

I If f is a function of both t and u, then ft denotes the derivative of fconsidered as a function of t alone, and fu denotes the derivative of fas a function of u alone.

I Example: If f (t, u) = t2 + 2tu + u3, then

ft(t, u) = 2t + 2u and fu(t, u) = 2t + 3u2.

I We call ft and fu the partial derivatives of f with respect to t and u,respectively.

I Notation: we also write

∂

∂tf (t, u) = ft(t, u) and

∂

∂uf (t, u) = fu(t, u).


Partial derivatives






∂


∂

∂uf (t, u) = fu(t, u).


Partial derivatives






∂


∂

∂uf (t, u) = fu(t, u).


Partial derivatives






∂


∂

∂uf (t, u) = fu(t, u).


Existence and uniqueness

Theorem

Suppose f is continuous on the open rectangle

R = {(t, u) : a < t < b, c < u < d}

and (t0, u0) is a point in R. Then the initial value problemdu

dt= f (t, u),

u(t0) = u0

has a solution u(t) on some interval (α, β) which contains t0. Moreover, iffu is continuous on R, then this solution is unique.


Interval of existence

I Note: The existence theorem does not tell us how large the interval(α, β) will be.

I LetR = {(t, u) : −∞ < t <∞,−∞ < u <∞} = R2.

I The initial value problem du

dt= 1− u2,

u(0) = 0

satisfies the conditions of the theorem on all of R.I The unique solution

u(t) =e2t − 1

e2t + 1

is defined on (−∞,∞).




I LetR = {(t, u) : −∞ < t <∞,−∞ < u <∞} = R2.


dt= 1− u2,

u(0) = 0


u(t) =e2t − 1

e2t + 1





I LetR = {(t, u) : −∞ < t <∞,−∞ < u <∞} = R2.


dt= 1− u2,

u(0) = 0

satisfies the conditions of the theorem on all of R.

I The unique solution

u(t) =e2t − 1

e2t + 1





I LetR = {(t, u) : −∞ < t <∞,−∞ < u <∞} = R2.


dt= 1− u2,

u(0) = 0


u(t) =e2t − 1

e2t + 1



Interval of existence (cont’d)

I Example continued:

I Similarly, the initial value problemdu

dt= 1 + u2,

u(0) = 0

satisfies the conditions of the theorem on all of R.I But its unique solution

u(t) = tan(t)

is defined on only(−π

2 ,π2

).



I Example continued:I Similarly, the initial value problem

du

dt= 1 + u2,

u(0) = 0

satisfies the conditions of the theorem on all of R.

I But its unique solutionu(t) = tan(t)


2 ,π2

).



I Example continued:I Similarly, the initial value problem

du

dt= 1 + u2,

u(0) = 0

satisfies the conditions of the theorem on all of R.I But its unique solution

u(t) = tan(t)


2 ,π2

).


Example

I Recall: Both u ≡ 0 and u = 127 t3 are solutions ofdu

dt= u

23 ,

u(0) = 0.

I Reason: Although f (t, u) = u23 is continuous on all of R2,

fu(t, u) =2

3u13

is not continuous at any points of the form (t, 0) (that is, on theu-axis).


Example

I Recall: Both u ≡ 0 and u = 127 t3 are solutions ofdu

dt= u

23 ,

u(0) = 0.

I Reason: Although f (t, u) = u23 is continuous on all of R2,

fu(t, u) =2

3u13

is not continuous at any points of the form (t, 0) (that is, on theu-axis).


Documents

Mathematics 22: Lecture 1 - Introduction - …math.furman.edu/~dcs/courses/math22/lectures/lecture-1.pdfCourse information I Course information is available at: dcs/dokuwiki/ Dan Sloughter