Mixing
Nonhomogeneous Second oder ODE
$begingroup$
I've recently started to learn ode, consider this equation $t''+at'-t=f$, where $a$ is not a constant and $f$ is a desired function.
$$y^{2} + ay-1= 0 quadLeftrightarrowquad y_{pm} = - frac{a}{2} pm sqrt{left(frac{a}{2}right)^{!2} +1}, $$
so for simplicity let's call it $y_{+}=A$ and $y_{-}=B $ as fundamental solution.
if I'm not mistaken this is the general solution:
$$t= frac{fA}{-BA'+AB'}-frac{fB}{-BA'+AB'}+C_{1}A+C_{2}B.$$
Now I want to know if this can be an explicit solution or not? And how can I show existence and uniqueness of this solution?
ordinary-differential-equations
edited Jan 22 at 15:56
Adrian Keister
5,27371933
asked Jan 22 at 15:49
AngelaAngela
61
$endgroup$
add a comment |
$begingroup$
I've recently started to learn ode, consider this equation $t''+at'-t=f$, where $a$ is not a constant and $f$ is a desired function.
$$y^{2} + ay-1= 0 quadLeftrightarrowquad y_{pm} = - frac{a}{2} pm sqrt{left(frac{a}{2}right)^{!2} +1}, $$
so for simplicity let's call it $y_{+}=A$ and $y_{-}=B $ as fundamental solution.
if I'm not mistaken this is the general solution:
$$t= frac{fA}{-BA'+AB'}-frac{fB}{-BA'+AB'}+C_{1}A+C_{2}B.$$
Now I want to know if this can be an explicit solution or not? And how can I show existence and uniqueness of this solution?
ordinary-differential-equations
edited Jan 22 at 15:56
Adrian Keister
5,27371933
asked Jan 22 at 15:49
AngelaAngela
61
$endgroup$
$begingroup$
The characteristic polynomial method only works for constant coefficients. (And the rest of the development is also wrong).
$endgroup$
– Yves Daoust
Jan 22 at 16:50
$begingroup$
@YvesDaoust So, what is the right solution for this equation? How can I solve it?
$endgroup$
– Angela
Jan 22 at 17:04
$begingroup$
In the general case, solving such a second order ODE equation is painful. As it is linear, you first solve the homogeneous part. Then you can try variation of constants to obtain a particular solution.
$endgroup$
– Yves Daoust
Jan 22 at 17:21
$begingroup$
@YvesDaoust and what is fundamental solution for homogeneous equation? It's $e^{y_{/pm}}t$
$endgroup$
– Angela
Jan 22 at 17:25
$begingroup$
No. Lookup "second order linear ODE with variable coefficients".
$endgroup$
– Yves Daoust
Jan 22 at 17:53
add a comment |
$begingroup$
I've recently started to learn ode, consider this equation $t''+at'-t=f$, where $a$ is not a constant and $f$ is a desired function.
$$y^{2} + ay-1= 0 quadLeftrightarrowquad y_{pm} = - frac{a}{2} pm sqrt{left(frac{a}{2}right)^{!2} +1}, $$
so for simplicity let's call it $y_{+}=A$ and $y_{-}=B $ as fundamental solution.
if I'm not mistaken this is the general solution:
$$t= frac{fA}{-BA'+AB'}-frac{fB}{-BA'+AB'}+C_{1}A+C_{2}B.$$
Now I want to know if this can be an explicit solution or not? And how can I show existence and uniqueness of this solution?
ordinary-differential-equations
edited Jan 22 at 15:56
Adrian Keister
5,27371933
asked Jan 22 at 15:49
AngelaAngela
61
$endgroup$
I've recently started to learn ode, consider this equation $t''+at'-t=f$, where $a$ is not a constant and $f$ is a desired function.
$$y^{2} + ay-1= 0 quadLeftrightarrowquad y_{pm} = - frac{a}{2} pm sqrt{left(frac{a}{2}right)^{!2} +1}, $$
so for simplicity let's call it $y_{+}=A$ and $y_{-}=B $ as fundamental solution.
if I'm not mistaken this is the general solution:
$$t= frac{fA}{-BA'+AB'}-frac{fB}{-BA'+AB'}+C_{1}A+C_{2}B.$$
Now I want to know if this can be an explicit solution or not? And how can I show existence and uniqueness of this solution?
ordinary-differential-equations
ordinary-differential-equations
edited Jan 22 at 15:56
Adrian Keister
5,27371933
asked Jan 22 at 15:49
AngelaAngela
61
edited Jan 22 at 15:56
Adrian Keister
5,27371933
asked Jan 22 at 15:49
AngelaAngela
61
edited Jan 22 at 15:56
Adrian Keister
5,27371933
edited Jan 22 at 15:56
Adrian Keister
5,27371933
edited Jan 22 at 15:56
Adrian Keister
5,27371933
5,27371933
asked Jan 22 at 15:49
AngelaAngela
61
asked Jan 22 at 15:49
AngelaAngela
61
asked Jan 22 at 15:49
AngelaAngela
61
61
$begingroup$
The characteristic polynomial method only works for constant coefficients. (And the rest of the development is also wrong).
$endgroup$
– Yves Daoust
Jan 22 at 16:50
$begingroup$
@YvesDaoust So, what is the right solution for this equation? How can I solve it?
$endgroup$
– Angela
Jan 22 at 17:04
$begingroup$
In the general case, solving such a second order ODE equation is painful. As it is linear, you first solve the homogeneous part. Then you can try variation of constants to obtain a particular solution.
$endgroup$
– Yves Daoust
Jan 22 at 17:21
$begingroup$
@YvesDaoust and what is fundamental solution for homogeneous equation? It's $e^{y_{/pm}}t$
$endgroup$
– Angela
Jan 22 at 17:25
$begingroup$
No. Lookup "second order linear ODE with variable coefficients".
$endgroup$
– Yves Daoust
Jan 22 at 17:53
add a comment |
$begingroup$
The characteristic polynomial method only works for constant coefficients. (And the rest of the development is also wrong).
$endgroup$
– Yves Daoust
Jan 22 at 16:50
$begingroup$
@YvesDaoust So, what is the right solution for this equation? How can I solve it?
$endgroup$
– Angela
Jan 22 at 17:04
$begingroup$
In the general case, solving such a second order ODE equation is painful. As it is linear, you first solve the homogeneous part. Then you can try variation of constants to obtain a particular solution.
$endgroup$
– Yves Daoust
Jan 22 at 17:21
$begingroup$
@YvesDaoust and what is fundamental solution for homogeneous equation? It's $e^{y_{/pm}}t$
$endgroup$
– Angela
Jan 22 at 17:25
$begingroup$
No. Lookup "second order linear ODE with variable coefficients".
$endgroup$
– Yves Daoust
Jan 22 at 17:53
$begingroup$
The characteristic polynomial method only works for constant coefficients. (And the rest of the development is also wrong).
$endgroup$
– Yves Daoust
Jan 22 at 16:50
$begingroup$
The characteristic polynomial method only works for constant coefficients. (And the rest of the development is also wrong).
$endgroup$
– Yves Daoust
Jan 22 at 16:50
$begingroup$
@YvesDaoust So, what is the right solution for this equation? How can I solve it?
$endgroup$
– Angela
Jan 22 at 17:04
$begingroup$
@YvesDaoust So, what is the right solution for this equation? How can I solve it?
$endgroup$
– Angela
Jan 22 at 17:04
$begingroup$
In the general case, solving such a second order ODE equation is painful. As it is linear, you first solve the homogeneous part. Then you can try variation of constants to obtain a particular solution.
$endgroup$
– Yves Daoust
Jan 22 at 17:21
$begingroup$
In the general case, solving such a second order ODE equation is painful. As it is linear, you first solve the homogeneous part. Then you can try variation of constants to obtain a particular solution.
$endgroup$
– Yves Daoust
Jan 22 at 17:21
$begingroup$
@YvesDaoust and what is fundamental solution for homogeneous equation? It's $e^{y_{/pm}}t$
$endgroup$
– Angela
Jan 22 at 17:25
$begingroup$
@YvesDaoust and what is fundamental solution for homogeneous equation? It's $e^{y_{/pm}}t$
$endgroup$
– Angela
Jan 22 at 17:25
$begingroup$
No. Lookup "second order linear ODE with variable coefficients".
$endgroup$
– Yves Daoust
Jan 22 at 17:53
$begingroup$
No. Lookup "second order linear ODE with variable coefficients".
$endgroup$
– Yves Daoust
Jan 22 at 17:53
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I would try a series solution - there isn't enough information on a, besides it being an arbitrary function.
answered Jan 23 at 2:53
FizzerFizzer
164
$endgroup$
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%2f3083325%2fnonhomogeneous-second-oder-ode%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
$begingroup$
I would try a series solution - there isn't enough information on a, besides it being an arbitrary function.
answered Jan 23 at 2:53
FizzerFizzer
164
$endgroup$
add a comment |
$begingroup$
I would try a series solution - there isn't enough information on a, besides it being an arbitrary function.
answered Jan 23 at 2:53
FizzerFizzer
164
$endgroup$
add a comment |
$begingroup$
I would try a series solution - there isn't enough information on a, besides it being an arbitrary function.
answered Jan 23 at 2:53
FizzerFizzer
164
$endgroup$
I would try a series solution - there isn't enough information on a, besides it being an arbitrary function.
answered Jan 23 at 2:53
FizzerFizzer
164
answered Jan 23 at 2:53
FizzerFizzer
164
answered Jan 23 at 2:53
FizzerFizzer
164
answered Jan 23 at 2:53
FizzerFizzer
164
164
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%2f3083325%2fnonhomogeneous-second-oder-ode%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
$begingroup$
The characteristic polynomial method only works for constant coefficients. (And the rest of the development is also wrong).
$endgroup$
– Yves Daoust
Jan 22 at 16:50
$begingroup$
@YvesDaoust So, what is the right solution for this equation? How can I solve it?
$endgroup$
– Angela
Jan 22 at 17:04
$begingroup$
In the general case, solving such a second order ODE equation is painful. As it is linear, you first solve the homogeneous part. Then you can try variation of constants to obtain a particular solution.
$endgroup$
– Yves Daoust
Jan 22 at 17:21
$begingroup$
@YvesDaoust and what is fundamental solution for homogeneous equation? It's $e^{y_{/pm}}t$
$endgroup$
– Angela
Jan 22 at 17:25
$begingroup$
No. Lookup "second order linear ODE with variable coefficients".
$endgroup$
– Yves Daoust
Jan 22 at 17:53