# 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 normal font. This means that you can
copy and paste all commands into *Mathematica*, change the parameters and
run them. You, as the user, are free to use the scripts for your needs to learn 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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# Existence and Uniqueness

** Theorem:** Suppose that

*f(x,y)*is a continuous function deﬁned in some rectangular region

*h*(possibly smaller than δ) so that a solution \( y = \phi (x) \) to the initial value problem

This theorem was proved in 1886 by the Italian mathematician Giuseppe Peano (1858--1932). Giuseppe Peano. Giuseppe Peano was a founder of symbolic logic whose interests centred on the foundations of mathematics and on the development of a formal logical language. In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891. In 1891, Peano started the Formulario Project. It was to be an "Encyclopedia of Mathematics", containing all known formulae and theorems of mathematical science using a standard notation invented by Peano.

In addition to his teaching at the University of Turin, Peano lectured at the Military Academy in Turin in 1886. The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations. However Émile Picard had independently discovered this method and had credited the German mathematician Hermann Schwarz (1843--1921) with discovering the method first. In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic.

** Theorem:** Let

*f(y)*be a continuous function on the closed interval [a,b] that has one null \( y^{\ast} \in (a,b) , \) namely, \( f(y^{\ast} ) =0 \) and \( f(y) \ne 0 \) for all other points \( y \in (a,b) . \) If the integral

** Theorem:** Suppose that

*f(x,y)*is uniformly Lipschitz continuous in

*y*(meaning the Lipschitz constant

*L*in the inequality \( |f(x,y_1 ) - f(x, y_2 )| \le L\,|y_1 - y_2 | \) can be taken independent of

*x*) and continuous in

*x*. Then, for some positive value δ there exists a unique solution \( y = \phi (x) \) to the initial value problem

** Theorem:** Let

*f(x,y)*be continuous for all

*(x,y)*in open rectangle \( R= \left\{ (x,y)\,:\, |x-x_0 | < a, \quad |y- y_0 | < b \,\right\} \) and Lipschitz continuous in

*y*, with constant

*L*independent of

*x*. Then there exists a unique solution to the initial value problem

*z(x)*is the solution to the same problem with the initial condition \( z(x_0 ) = z_0 , \) then

** Theorem:** Suppose that

*f(x,y)*and \( \frac{\partial f}{\partial y} \) are continuous functions deﬁned in some rectangular region

*R*:

*M*and

*K*, then the initial value problem

*f(x,y)*and the solution

*y(x)*are continuous functions)

** Corollary:** The continuous dependence of the solutions on the initial conditions holds whenever slope function

*f*satisfies a global Lipschitz condition.

** Corollary:** If the solution

*y(x)*of the initial value problem \( y' = f(x,y), \ y(x_0 )= y_0 \) has an a priori bound

*M*, i.e., \( |y(x)| \le M \) whenever

*y(x)*exists, then the solution exists for all \( x \in \mathbb{R} . \)

**Example**.
Consider the initial value problem

*C*is an arbitrary constant. Since we consider only positive branch of the square root function, the above formula is valid only when \( x \ge C . \) Therefore, we get a family of solutions (which is also called the general solution) depending on a parameter

*C*:

*Mathematica*, we plot some solutions

q2 = Plot[y = 0, {x, -3.5, 3.5}, PlotStyle -> {Thick, Black}] (* singular solution *)

graph4[CC_] :=

Module[{}, Plot[Evaluate[q[x, CC]], {x, -3.5, 3.5}, AxesLabel -> {x, y},

PlotRange -> {{-3.5, 3.5}, {-0.5, 6}}, AspectRatio -> 1, DisplayFunction -> Identity,

PlotStyle -> RGBColor[1, 0, 0]]]

initlist = {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, -1, -2, -3, -4};

Module[{i, newgraph}, graphlist = {}; Do[CC = initlist[[i]];

newgraph = graph4[CC];

graphlist = Append[graphlist, newgraph], {i, 1, Length[initlist]}]]

solgraph =

Show[q2, graphlist, {PlotStyle -> {Black, Thick}, {DisplayFunction -> $DisplayFunction}}]

In this sequence of command, I am first entering the family of solutions to the differential equation. Since using **C** is prohibited in *Mathematica*, we use *CC* instead. Then we use two subroutines, one for plotting solutions, and another one for looping with respect to constant *C*, Finally, we display all graphs.

We can also check that the given initial value problem has multiple solutions by evaluating integral

**Example**.
Consider the initial value problem for the Riccati equation

# Plotting Solutions to ODEs

# Direction Fields

# Separable Equations

# Equations Reducible to the Separable Equations

# Equations with Linear Fractions

# Exact Equations

# Integrating Factors

# Linear Equations

# Bernoulli Equations

# Riccati Equations

# Existence and Uniqueness

# Qualitative Analysis

# Orthogonal Trajectories

# Population Models

# Applications

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