Preface

This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all comamnds into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts to your needs for learning how to use the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

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Orthogonal Trajectories

Example. (Orthogonal trajectories) Given the family of curves x*y=c, where c is a constant. These are hyperbolas in quadrants 1 and 3 for c>0 or quadrants 2 and 4 for c<0, having the axes as asymptotes. They are rectengular hyperbolas because their asymptotes are perpendicular. The differential equation of the family is x y' + y =0, or dy/dx = -y/x . The orthogonality condition says that the equation of the orthogonal curves should be - dx/dy = -y/x or dy/dx = y/x . The latter has the general solution (after separation of variables): y^2 = x^2 +c. Now we plot these curves.

a := ContourPlot[{x y == 1, x y == 0.2, x y == 0.5, x y == 1.5,
x y == -1, x y == -0.2, x y == -0.5, x y == -1.5}, {x, -2, 2}, {y, -2, 2}]
b := ContourPlot[{x^2 == y^2 + 1, x^2 == y^2 + 0.1, x^2 == y^2 + 1.9,
x^2 == y^2 - 1, x^2 == y^2 - 0.1, x^2 == y^2 - 1.9}, {x, -2, 2}, {y, -2, 2}]
Show[a, b]



Example 2.11.2.
The differential equation Exp[y] dy = Sin[x] dx defines its
family of solutions: Exp[y] == Sin {x] +c .
The orthiogonal trajectories are ln | Tan[x/2] = Exp[y] +c.

a := ContourPlot[{Exp[y] + Cos[x] == 3, Exp[y] + Cos[x] == 0.2,
Exp[y] + Cos[x] == 2, Exp[y] + Cos[x] == 0.5,
Cos[x] + Exp[y] == 1.5, Cos[x] + Exp[ y] == -2,
Cos[x] + Exp[y] == -0.2, Cos[x] + Exp[ y] == -0.5,
Cos[x] + Exp[y] == -1.5}, {x, -2, 2}, {y, -2, 2}]
b := ContourPlot[{Log[Tan[x]] == Exp[-y] + 1,
Log[Tan[x]] == Exp[-y] + 0.1, Log[Tan[x]] == Exp[-y] + 0.9,
Log[Tan[x]] == Exp[-y] - 1, x Log[Tan[x]] == Exp[-y] - 0.1,
Log[Tan[x]] == Exp[-y] - 3.9}, {x, -2, 2}, {y, -2, 2}]
Show[a, b]
>>>>> ortho2.jpg

 

 

First Order ODEs

Plotting Solutions to ODEs

Direction Fields

Separable Equations

Equations Reducible to the Separable Equations

Equations with Linear Fractions

Exact Equations

Integrating Factors

Linear Equations

RC circuits

Bernoulli Equations

Riccati Equations

Existence and Uniqueness

Qualitative Analysis

Bifurcations

Orthogonal Trajectories

Population Models

Applications

 

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