On the existence and uniqueness of solutions of non-linear differential equation system
$begingroup$
Are there theorems that could be applied to show the existence/uniqueness of solution of such system :
$$
(S) left { begin{aligned}
p''&= f_1(b) \
q'' &= f_2(x,p,q,b)\
q~~ &= f_3(x,p,b)
end{aligned}
right.
$$
where $f_1$, $f_2$ and $f_3$ are non linear functions ? and $p$, $b$ and $q$ are real and positive functions defined on a compact $[a,b] subset mathbb{R+}$ ?
differential-equations nonlinear-system
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
$endgroup$
add a comment |
$begingroup$
Are there theorems that could be applied to show the existence/uniqueness of solution of such system :
$$
(S) left { begin{aligned}
p''&= f_1(b) \
q'' &= f_2(x,p,q,b)\
q~~ &= f_3(x,p,b)
end{aligned}
right.
$$
where $f_1$, $f_2$ and $f_3$ are non linear functions ? and $p$, $b$ and $q$ are real and positive functions defined on a compact $[a,b] subset mathbb{R+}$ ?
differential-equations nonlinear-system
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
$endgroup$
$begingroup$
Is the last line of $(S)$ supposed to be $q=...$?
$endgroup$
– Not Mike
Dec 4 '18 at 17:11
$begingroup$
Yes why ? it characterizes how q is a function of the other unknowns $p$ and $b$.
$endgroup$
– AlexC75
Dec 4 '18 at 17:12
1
$begingroup$
If or where the last equation can be solved for $b$ this is an index-1 DAE in semi-explicit form. The general theory of differential-algebraic equations applies, which gives here apart from said solvability no further restrictions.
$endgroup$
– LutzL
Dec 4 '18 at 17:21
add a comment |
$begingroup$
Are there theorems that could be applied to show the existence/uniqueness of solution of such system :
$$
(S) left { begin{aligned}
p''&= f_1(b) \
q'' &= f_2(x,p,q,b)\
q~~ &= f_3(x,p,b)
end{aligned}
right.
$$
where $f_1$, $f_2$ and $f_3$ are non linear functions ? and $p$, $b$ and $q$ are real and positive functions defined on a compact $[a,b] subset mathbb{R+}$ ?
differential-equations nonlinear-system
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
$endgroup$
Are there theorems that could be applied to show the existence/uniqueness of solution of such system :
$$
(S) left { begin{aligned}
p''&= f_1(b) \
q'' &= f_2(x,p,q,b)\
q~~ &= f_3(x,p,b)
end{aligned}
right.
$$
where $f_1$, $f_2$ and $f_3$ are non linear functions ? and $p$, $b$ and $q$ are real and positive functions defined on a compact $[a,b] subset mathbb{R+}$ ?
differential-equations nonlinear-system
differential-equations nonlinear-system
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
edited Dec 4 '18 at 17:16
LutzL
56.7k42054
56.7k42054
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
asked Dec 4 '18 at 17:00
AlexC75AlexC75
216
216
$begingroup$
Is the last line of $(S)$ supposed to be $q=...$?
$endgroup$
– Not Mike
Dec 4 '18 at 17:11
$begingroup$
Yes why ? it characterizes how q is a function of the other unknowns $p$ and $b$.
$endgroup$
– AlexC75
Dec 4 '18 at 17:12
1
$begingroup$
If or where the last equation can be solved for $b$ this is an index-1 DAE in semi-explicit form. The general theory of differential-algebraic equations applies, which gives here apart from said solvability no further restrictions.
$endgroup$
– LutzL
Dec 4 '18 at 17:21
add a comment |
$begingroup$
Is the last line of $(S)$ supposed to be $q=...$?
$endgroup$
– Not Mike
Dec 4 '18 at 17:11
$begingroup$
Yes why ? it characterizes how q is a function of the other unknowns $p$ and $b$.
$endgroup$
– AlexC75
Dec 4 '18 at 17:12
1
$begingroup$
If or where the last equation can be solved for $b$ this is an index-1 DAE in semi-explicit form. The general theory of differential-algebraic equations applies, which gives here apart from said solvability no further restrictions.
$endgroup$
– LutzL
Dec 4 '18 at 17:21
$begingroup$
Is the last line of $(S)$ supposed to be $q=...$?
$endgroup$
– Not Mike
Dec 4 '18 at 17:11
$begingroup$
Is the last line of $(S)$ supposed to be $q=...$?
$endgroup$
– Not Mike
Dec 4 '18 at 17:11
$begingroup$
Yes why ? it characterizes how q is a function of the other unknowns $p$ and $b$.
$endgroup$
– AlexC75
Dec 4 '18 at 17:12
$begingroup$
Yes why ? it characterizes how q is a function of the other unknowns $p$ and $b$.
$endgroup$
– AlexC75
Dec 4 '18 at 17:12
1
1
$begingroup$
If or where the last equation can be solved for $b$ this is an index-1 DAE in semi-explicit form. The general theory of differential-algebraic equations applies, which gives here apart from said solvability no further restrictions.
$endgroup$
– LutzL
Dec 4 '18 at 17:21
$begingroup$
If or where the last equation can be solved for $b$ this is an index-1 DAE in semi-explicit form. The general theory of differential-algebraic equations applies, which gives here apart from said solvability no further restrictions.
$endgroup$
– LutzL
Dec 4 '18 at 17:21
add a comment |
0
active
oldest
votes
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%2f3025829%2fon-the-existence-and-uniqueness-of-solutions-of-non-linear-differential-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3025829%2fon-the-existence-and-uniqueness-of-solutions-of-non-linear-differential-equation%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
$begingroup$
Is the last line of $(S)$ supposed to be $q=...$?
$endgroup$
– Not Mike
Dec 4 '18 at 17:11
$begingroup$
Yes why ? it characterizes how q is a function of the other unknowns $p$ and $b$.
$endgroup$
– AlexC75
Dec 4 '18 at 17:12
1
$begingroup$
If or where the last equation can be solved for $b$ this is an index-1 DAE in semi-explicit form. The general theory of differential-algebraic equations applies, which gives here apart from said solvability no further restrictions.
$endgroup$
– LutzL
Dec 4 '18 at 17:21