Applying the chain rule for a multivariate function
$begingroup$
Suppose we have the system
$$ z_k' = f_k(x,z_1,...,z_n) $$
of first-order ODE. I want to differentiate this equation j times. IF we differentiate with respect to $x$ we have
$$ z_1'' = frac{ d }{dx} f_1(x,z_1,...,z_n) $$
by chain rule we have that RHS is
$$ frac{ partial f_1 }{partial x } x' + frac{ partial f_1 }{partial z_1} z_1' + ... + frac{ partial f_1 }{partial z_n } z_n' = f_1^{(1)}(x,z_1,...,z_n)$$
Now the jth derivative will also give a function of x and the $z's$ say $f^{(j-1)}_j (x,z_1,...,z_n)$
Now, my book says that if we consider the n equations $f_j^{j-1} = z_1^{(j)}$ then we can solve to obtain an equation of the form
$$ p_0 y + p_1 y'' + p_2 y''' + ... + y^{(n)} = f(x) $$
How is this possible?
ordinary-differential-equations
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
$endgroup$
add a comment |
$begingroup$
Suppose we have the system
$$ z_k' = f_k(x,z_1,...,z_n) $$
of first-order ODE. I want to differentiate this equation j times. IF we differentiate with respect to $x$ we have
$$ z_1'' = frac{ d }{dx} f_1(x,z_1,...,z_n) $$
by chain rule we have that RHS is
$$ frac{ partial f_1 }{partial x } x' + frac{ partial f_1 }{partial z_1} z_1' + ... + frac{ partial f_1 }{partial z_n } z_n' = f_1^{(1)}(x,z_1,...,z_n)$$
Now the jth derivative will also give a function of x and the $z's$ say $f^{(j-1)}_j (x,z_1,...,z_n)$
Now, my book says that if we consider the n equations $f_j^{j-1} = z_1^{(j)}$ then we can solve to obtain an equation of the form
$$ p_0 y + p_1 y'' + p_2 y''' + ... + y^{(n)} = f(x) $$
How is this possible?
ordinary-differential-equations
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
$endgroup$
add a comment |
$begingroup$
Suppose we have the system
$$ z_k' = f_k(x,z_1,...,z_n) $$
of first-order ODE. I want to differentiate this equation j times. IF we differentiate with respect to $x$ we have
$$ z_1'' = frac{ d }{dx} f_1(x,z_1,...,z_n) $$
by chain rule we have that RHS is
$$ frac{ partial f_1 }{partial x } x' + frac{ partial f_1 }{partial z_1} z_1' + ... + frac{ partial f_1 }{partial z_n } z_n' = f_1^{(1)}(x,z_1,...,z_n)$$
Now the jth derivative will also give a function of x and the $z's$ say $f^{(j-1)}_j (x,z_1,...,z_n)$
Now, my book says that if we consider the n equations $f_j^{j-1} = z_1^{(j)}$ then we can solve to obtain an equation of the form
$$ p_0 y + p_1 y'' + p_2 y''' + ... + y^{(n)} = f(x) $$
How is this possible?
ordinary-differential-equations
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
$endgroup$
Suppose we have the system
$$ z_k' = f_k(x,z_1,...,z_n) $$
of first-order ODE. I want to differentiate this equation j times. IF we differentiate with respect to $x$ we have
$$ z_1'' = frac{ d }{dx} f_1(x,z_1,...,z_n) $$
by chain rule we have that RHS is
$$ frac{ partial f_1 }{partial x } x' + frac{ partial f_1 }{partial z_1} z_1' + ... + frac{ partial f_1 }{partial z_n } z_n' = f_1^{(1)}(x,z_1,...,z_n)$$
Now the jth derivative will also give a function of x and the $z's$ say $f^{(j-1)}_j (x,z_1,...,z_n)$
Now, my book says that if we consider the n equations $f_j^{j-1} = z_1^{(j)}$ then we can solve to obtain an equation of the form
$$ p_0 y + p_1 y'' + p_2 y''' + ... + y^{(n)} = f(x) $$
How is this possible?
ordinary-differential-equations
ordinary-differential-equations
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
asked Dec 25 '18 at 18:14
NeymarNeymar
406214
406214
add a comment |
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%2f3052326%2fapplying-the-chain-rule-for-a-multivariate-function%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%2f3052326%2fapplying-the-chain-rule-for-a-multivariate-function%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
