Steps used to solve this riccati differential equation?
$begingroup$
I am reading through examples of linear filtering problems for SDE's, and the process first requires solving a (deterministic) riccati differential equation. In one of the examples, this is given by:
$$frac{dS_t}{dt} = frac{-1}{m^2}S^2_t + c^2$$
to solve it, they first rearrange it as
$$frac{m^2 dS_t}{m^2 c^2 -S^2_t} = dt$$
and then state (without working or explanation) that this gives
$$left| frac{mc+S_t}{mc-S_t} right|= left| frac{mc+a^2}{mc-a^2} right| exp left( frac{2ct}{m} right)$$
And I'm unsure how to get from one to the other.
Clearly, since $t$ has emerged, it involves integrating both sides. But besides that I'm lost. Could someone explain this or give some steps/outline for me to do myself?
ordinary-differential-equations self-learning
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
$endgroup$
add a comment |
$begingroup$
I am reading through examples of linear filtering problems for SDE's, and the process first requires solving a (deterministic) riccati differential equation. In one of the examples, this is given by:
$$frac{dS_t}{dt} = frac{-1}{m^2}S^2_t + c^2$$
to solve it, they first rearrange it as
$$frac{m^2 dS_t}{m^2 c^2 -S^2_t} = dt$$
and then state (without working or explanation) that this gives
$$left| frac{mc+S_t}{mc-S_t} right|= left| frac{mc+a^2}{mc-a^2} right| exp left( frac{2ct}{m} right)$$
And I'm unsure how to get from one to the other.
Clearly, since $t$ has emerged, it involves integrating both sides. But besides that I'm lost. Could someone explain this or give some steps/outline for me to do myself?
ordinary-differential-equations self-learning
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
$endgroup$
add a comment |
$begingroup$
I am reading through examples of linear filtering problems for SDE's, and the process first requires solving a (deterministic) riccati differential equation. In one of the examples, this is given by:
$$frac{dS_t}{dt} = frac{-1}{m^2}S^2_t + c^2$$
to solve it, they first rearrange it as
$$frac{m^2 dS_t}{m^2 c^2 -S^2_t} = dt$$
and then state (without working or explanation) that this gives
$$left| frac{mc+S_t}{mc-S_t} right|= left| frac{mc+a^2}{mc-a^2} right| exp left( frac{2ct}{m} right)$$
And I'm unsure how to get from one to the other.
Clearly, since $t$ has emerged, it involves integrating both sides. But besides that I'm lost. Could someone explain this or give some steps/outline for me to do myself?
ordinary-differential-equations self-learning
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
$endgroup$
I am reading through examples of linear filtering problems for SDE's, and the process first requires solving a (deterministic) riccati differential equation. In one of the examples, this is given by:
$$frac{dS_t}{dt} = frac{-1}{m^2}S^2_t + c^2$$
to solve it, they first rearrange it as
$$frac{m^2 dS_t}{m^2 c^2 -S^2_t} = dt$$
and then state (without working or explanation) that this gives
$$left| frac{mc+S_t}{mc-S_t} right|= left| frac{mc+a^2}{mc-a^2} right| exp left( frac{2ct}{m} right)$$
And I'm unsure how to get from one to the other.
Clearly, since $t$ has emerged, it involves integrating both sides. But besides that I'm lost. Could someone explain this or give some steps/outline for me to do myself?
ordinary-differential-equations self-learning
ordinary-differential-equations self-learning
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
asked Dec 22 '18 at 6:15
XiaomiXiaomi
1,066115
1,066115
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Don't see this as Riccati equation, see it as separable ODE with constant solutions at $pm mc$. Then the last form has a partial fraction decomposition
$$
frac{mdS}{S+mc}-frac{mdS}{S-mc} = 2cdt
$$
which after integration and determination of the integration constant should give your solution form.
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
$endgroup$
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%2f3049162%2fsteps-used-to-solve-this-riccati-differential-equation%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
$begingroup$
Don't see this as Riccati equation, see it as separable ODE with constant solutions at $pm mc$. Then the last form has a partial fraction decomposition
$$
frac{mdS}{S+mc}-frac{mdS}{S-mc} = 2cdt
$$
which after integration and determination of the integration constant should give your solution form.
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
$endgroup$
add a comment |
$begingroup$
Don't see this as Riccati equation, see it as separable ODE with constant solutions at $pm mc$. Then the last form has a partial fraction decomposition
$$
frac{mdS}{S+mc}-frac{mdS}{S-mc} = 2cdt
$$
which after integration and determination of the integration constant should give your solution form.
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
$endgroup$
add a comment |
$begingroup$
Don't see this as Riccati equation, see it as separable ODE with constant solutions at $pm mc$. Then the last form has a partial fraction decomposition
$$
frac{mdS}{S+mc}-frac{mdS}{S-mc} = 2cdt
$$
which after integration and determination of the integration constant should give your solution form.
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
$endgroup$
Don't see this as Riccati equation, see it as separable ODE with constant solutions at $pm mc$. Then the last form has a partial fraction decomposition
$$
frac{mdS}{S+mc}-frac{mdS}{S-mc} = 2cdt
$$
which after integration and determination of the integration constant should give your solution form.
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
answered Dec 22 '18 at 7:06
LutzLLutzL
59.3k42057
59.3k42057
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%2f3049162%2fsteps-used-to-solve-this-riccati-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
