Solutions of certain differential equations in two variablesProve theorem of Existence and Uniqueness of solutions of a $n$-th order differential equation using Picard-Lindelöf theoremDisplacement of differential equationfinding a solution of an autonomous differential equation with two variablesnecessary and sufficient conditions for the existence of solutions.Is it the Picard-Lindelöf Theorem?Is there a harder way to prove that if an integral curve's derivative vanishes then the curve is constant?Boundedness of solutions to a certain system of “weakly-coupled” linear differential equationsEquivalence of two definitions for submanifoldsProperty of solutions of differential equationSmooth map with null differential at each point are constant on the connected component of the domain
Print name if parameter passed to function
Hide Select Output from T-SQL
Is the destination of a commercial flight important for the pilot?
Is there a good way to store credentials outside of a password manager?
Is exact Kanji stroke length important?
Why Were Madagascar and New Zealand Discovered So Late?
Can a monster with multiattack use this ability if they are missing a limb?
voltage of sounds of mp3files
Valid Badminton Score?
Student evaluations of teaching assistants
There is only s̶i̶x̶t̶y one place he can be
Everything Bob says is false. How does he get people to trust him?
How does residential electricity work?
How will losing mobility of one hand affect my career as a programmer?
Applicability of Single Responsibility Principle
Can somebody explain Brexit in a few child-proof sentences?
Tiptoe or tiphoof? Adjusting words to better fit fantasy races
Curses work by shouting - How to avoid collateral damage?
Implement the Thanos sorting algorithm
Will it be accepted, if there is no ''Main Character" stereotype?
Modify casing of marked letters
The plural of 'stomach"
Efficiently merge handle parallel feature branches in SFDX
Why does John Bercow say “unlock” after reading out the results of a vote?
Solutions of certain differential equations in two variables
Prove theorem of Existence and Uniqueness of solutions of a $n$-th order differential equation using Picard-Lindelöf theoremDisplacement of differential equationfinding a solution of an autonomous differential equation with two variablesnecessary and sufficient conditions for the existence of solutions.Is it the Picard-Lindelöf Theorem?Is there a harder way to prove that if an integral curve's derivative vanishes then the curve is constant?Boundedness of solutions to a certain system of “weakly-coupled” linear differential equationsEquivalence of two definitions for submanifoldsProperty of solutions of differential equationSmooth map with null differential at each point are constant on the connected component of the domain
$begingroup$
Let $phi:[a,b] to mathbbR^n$ be a smooth function such that $phi(a)=phi(b)=0,$ and let $x,y in mathbbR^n$ be arbitrary points. Suppose we have another fixed smooth path $f: [a,b] to mathbbR^n$.
How do I prove that there always exists a smooth $psi:mathbbR times [a,b] to mathbbR^n$ such that:
$$psi(0,t)= f(t), psi(s,a)=x, psi(s,b)=y$$ and $$fracpartialpartial s psi(0,t)= phi(t)?$$
I would assume to use some version of Picard-Lindelöf, but I do not see how to apply it here.
ordinary-differential-equations differential-geometry
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
$endgroup$
add a comment |
$begingroup$
Let $phi:[a,b] to mathbbR^n$ be a smooth function such that $phi(a)=phi(b)=0,$ and let $x,y in mathbbR^n$ be arbitrary points. Suppose we have another fixed smooth path $f: [a,b] to mathbbR^n$.
How do I prove that there always exists a smooth $psi:mathbbR times [a,b] to mathbbR^n$ such that:
$$psi(0,t)= f(t), psi(s,a)=x, psi(s,b)=y$$ and $$fracpartialpartial s psi(0,t)= phi(t)?$$
I would assume to use some version of Picard-Lindelöf, but I do not see how to apply it here.
ordinary-differential-equations differential-geometry
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
$endgroup$
add a comment |
$begingroup$
Let $phi:[a,b] to mathbbR^n$ be a smooth function such that $phi(a)=phi(b)=0,$ and let $x,y in mathbbR^n$ be arbitrary points. Suppose we have another fixed smooth path $f: [a,b] to mathbbR^n$.
How do I prove that there always exists a smooth $psi:mathbbR times [a,b] to mathbbR^n$ such that:
$$psi(0,t)= f(t), psi(s,a)=x, psi(s,b)=y$$ and $$fracpartialpartial s psi(0,t)= phi(t)?$$
I would assume to use some version of Picard-Lindelöf, but I do not see how to apply it here.
ordinary-differential-equations differential-geometry
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
$endgroup$
Let $phi:[a,b] to mathbbR^n$ be a smooth function such that $phi(a)=phi(b)=0,$ and let $x,y in mathbbR^n$ be arbitrary points. Suppose we have another fixed smooth path $f: [a,b] to mathbbR^n$.
How do I prove that there always exists a smooth $psi:mathbbR times [a,b] to mathbbR^n$ such that:
$$psi(0,t)= f(t), psi(s,a)=x, psi(s,b)=y$$ and $$fracpartialpartial s psi(0,t)= phi(t)?$$
I would assume to use some version of Picard-Lindelöf, but I do not see how to apply it here.
ordinary-differential-equations differential-geometry
ordinary-differential-equations differential-geometry
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
asked Mar 17 at 13:04
inequalitynoob2inequalitynoob2
1447
1447
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is not a differential equation. A differential equation would relate the derivative(s) of $psi(s,t)$ to $s$, $t$, and $psi(s,t)$ throughout the domain. Your equations are essentially just boundary conditions and initial conditions for $psi$.
In order to find such a function, you have to stipulate first that $f(a)=x$ and $f(b)=y$; otherwise the conditions are self-contradictory.
With this assumption, there are many such functions, but here's a simple one that you can write down explicitly:
$$
psi(s,t) = f(t) + sphi(t).
$$
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3151520%2fsolutions-of-certain-differential-equations-in-two-variables%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is not a differential equation. A differential equation would relate the derivative(s) of $psi(s,t)$ to $s$, $t$, and $psi(s,t)$ throughout the domain. Your equations are essentially just boundary conditions and initial conditions for $psi$.
In order to find such a function, you have to stipulate first that $f(a)=x$ and $f(b)=y$; otherwise the conditions are self-contradictory.
With this assumption, there are many such functions, but here's a simple one that you can write down explicitly:
$$
psi(s,t) = f(t) + sphi(t).
$$
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
$endgroup$
add a comment |
$begingroup$
This is not a differential equation. A differential equation would relate the derivative(s) of $psi(s,t)$ to $s$, $t$, and $psi(s,t)$ throughout the domain. Your equations are essentially just boundary conditions and initial conditions for $psi$.
In order to find such a function, you have to stipulate first that $f(a)=x$ and $f(b)=y$; otherwise the conditions are self-contradictory.
With this assumption, there are many such functions, but here's a simple one that you can write down explicitly:
$$
psi(s,t) = f(t) + sphi(t).
$$
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
$endgroup$
add a comment |
$begingroup$
This is not a differential equation. A differential equation would relate the derivative(s) of $psi(s,t)$ to $s$, $t$, and $psi(s,t)$ throughout the domain. Your equations are essentially just boundary conditions and initial conditions for $psi$.
In order to find such a function, you have to stipulate first that $f(a)=x$ and $f(b)=y$; otherwise the conditions are self-contradictory.
With this assumption, there are many such functions, but here's a simple one that you can write down explicitly:
$$
psi(s,t) = f(t) + sphi(t).
$$
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
$endgroup$
This is not a differential equation. A differential equation would relate the derivative(s) of $psi(s,t)$ to $s$, $t$, and $psi(s,t)$ throughout the domain. Your equations are essentially just boundary conditions and initial conditions for $psi$.
In order to find such a function, you have to stipulate first that $f(a)=x$ and $f(b)=y$; otherwise the conditions are self-contradictory.
With this assumption, there are many such functions, but here's a simple one that you can write down explicitly:
$$
psi(s,t) = f(t) + sphi(t).
$$
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
answered Mar 17 at 17:44
Jack LeeJack Lee
27.7k54868
27.7k54868
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3151520%2fsolutions-of-certain-differential-equations-in-two-variables%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
