Title: Theory / Unimportant / Math / Analysis / ODE+14
Tags: LaTeXError hasSeenConjDef

Name: A region
Definition: path-connected
Variables: \(T\) a topological space
Definition2: open
Conjdef: A subset of \(T\)
Definition3: Non-empty
Name2: A domain in \(T\)

Name: A bounded region
Definition: A region contained in a circle
Page: 14
Chapter: 2
Section: C
Variables: \(T\) a topological space
Index: 68
Kind: Def

Name: The implicit function of \(x\) defined by \(f(x,y)=0\)
Definition: A function \(g:I\to\mathbb R\) such that \(f(x,g(x))=0\)
Page: 16b
Chapter: 2
Section: D
Variables: \(f(x,y):\mathbb R^2\to\mathbb R\)
\(I\subseteq \mathbb R\) an interval
Index: 81
Kind: Def
Assuming: a unique such function exists

Name: Elementary functions
Definition: containing \(x\mapsto x^{p/q}\) with \(p,q\ge 0\)
Page: 17
Chapter: 2
Section: E
Index: 68
Kind: Def
Definition2: exponential and logarithm
Conjdef: the set
Definition3: trigonometric functions and their inverses
Definition4: closed under composition
Definition5: closed under the four arithmetic operations

Name: An ordinary differential equation
Definition: An equation involving \(x\), \(f(x)\) and at least one of its derivatives
Page: 20
Chapter: 3
Section: E
Variables: \(I\) a bounded interval
\(f(x):I\to\mathbb R\)
Index: 1
Kind: Def
Abbreviation: ODE
Denoted by: \(g(x,y)=0\), where \(y\) denotes \(f(x)\)

Name: The order of the ODE \(g\)
Definition: The order of the highest derivative involved in the equation \(g\)
Page: 21
Chapter: 3
Section: B
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE in \(x\)

Name: A solution of \(g\)
Definition: A function \(f:I\to\mathbb R\) such that [$]\defFun{I}{\mathbb R}{x}{g(x,f(x))}[/$] is uniformly zero
Page: 21
Chapter: 3
Section: B
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE
Name2: An explicit solution of \(g\)

Question: How to test whether \(f\) is a solution of \(g\) ?
Answer: Replace [$]\dv[n]{y}{x}[/$] by \(f^{(n)}\) in \(g\), and compute that \(g\) is uniformly \(0\)
Page: 21
Chapter: 3
Section: B
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE
\(f(x):I\to\mathbb R\)

Question: How to test whether \(f\) is an implicit solution of \(g\) ?
Answer: differentiate \(f\), checks whether:
-[$]\dv{f}{x}=g[/$]
-\(f\) defines an implicit function on \(I\)
Page: 25
Chapter: 3
Section: B
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE
\(f(x,y):I\times\mathbb R\to\mathbb R\) an implicity function
Index: 651
Kind: Comment

Name: An implicit solution of \(g\)
Definition: An implicit function \(f(x,y):I\times\mathbb R\to\mathbb R\) defining a function \(h:I\to\mathbb R\) such that \(h\) is a solution of \(g\)
Page: 21
Chapter: 3
Section: B
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE

Question: exemple of \(n\)-th order differential equation with a single solution
Answer: \((y^{(n)})^2+y^2=0\) or generally \(f(y^{(n)})+f(y)\) for any function \(f:\mathbb R\to\mathbb R^\ge0\) such that \(f^{-1}(0)=\{0\}\)
Page: 29
Chapter: 4
Section: A

Question: exemple of \(n\)-th order differential equation without a solution
Answer: \((y^{(n)})^2+1=0\)
Page: 29
Chapter: 4
Section: A

Name: An \(n\)-parameter family of solutions
Definition: A function \(f(c_1,\dots,c_n,x)\) such that, for each \((c_1,\dots,c_n)\) fixed, the function \(x\mapsto f(c_1,\dots,c_n,x)\) is a solution of \(g\)
Page: 30
Chapter: 4
Section: A
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE
Name2: A general solution of \(g\) (to be avoided in this book)

Question: How to find an OGE whose solution is \(f\) ?
Answer: differentiate \(f\) \(n\) times, check whether you can find a way to combine those \(n\) functions to obtain an expression equivalent to a function without those parameters. Return this expression, where each \(i\)-th derivative is in the form \(y^{(i)}\)
Page: 32
Chapter: 4
Section: B
Variables: \(I\) a bounded interval
\(f(c_1,\dots,c_n,x)\) a \(n\)-parameter family of solution

Name: A particular solution of \(g\) (to be avoided)
Definition: Some \(x\mapsto f(c_1,\dots,c_n,x)\) for \((c_1,\dots,c_n)\) fixed
Page: 32
Chapter: 4
Section: C
Variables: \(I\) a bounded interval
\(g(x,y)\) an OGE
\(f(c_1,\dots,c_n,x)\) a \(n\)-parameter family of solution of \(g\)

Name: A singular solution of \(g\) (to be avoided)
Definition: A solution of \(g\) which can not be obtained from a \(n\)-parameter family of solution
Page: 32
Chapter: 4
Section: C
Variables: \(I\) a bounded interval
\(g(x,y)\) an OGE

Question: Why should we avoid the notion of singular solution of \(g\) ?
Answer: Because a solution may or may not belong to a family of solution, depending on how the family is presented.
Page: 5
Chapter: 4
Section: C

Question: Example of two families of functions whose particular functions are the same, except for two points
Answer: \(y=\frac{1}{(x+c)^2}\) and \(y=\frac{C^2}{(Cx+1)^2}\) are related by taking \(c=\frac 1C\)
Page: 32
Chapter: 4
Section: C
Variables: \(I\) a bounded interval

Name: A general solution of \(x\)
Definition: A \(n\)-paramater family of solution \(f(c_1,\dots,c_n,x)\) such that every solutions is obtained by fixing the \(c_i\)'s to some value
Page: 35
Chapter: 4
Section: C
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE

Name: Initial condition for \(g\)
Definition: \(n\) conditions which enable to determine the values of the \(c_i\)'s
Page: 36
Chapter: 4
Section: C
Variables: \(I\) a bounded interval
\(g(x,y)\) an ODE of degree \(n\)
\(f(c_1,\dots,c_n,x)\) a \(n\) parameter family of solution
Index: 71
Kind: Def