I need some help finding a second solution to $t^2 y''-2ty=3t^2-1$ given the fact that $y_1=t^{-1}$
Using the fact that $y_1$=$t^{-1}$ then $y_2$= z $cdot$ $t^{-1}$
$y_2'= z'cdot t^{-1} - z cdot t^{-2}$
$y_2''= z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2 cdot z cdot t^{-3}$
Substituting into the original equation:
$t^2 (z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2cdot z cdot t^{-3})-2t(z cdot t^{-1})$
Which I then simplified into:
$tcdot z''-4cdot z'+2zcdot t^{-1} -2z = 3 t^2 -1$
I want to use the method of reduction of order to find the second solution, by setting $w=z'$, but there are too many orders of $z$ remaining in the equation. Not sure where I keep going wrong!
differential-equations reduction-of-order-ode
edited Dec 12 '18 at 8:04
postmortes
1,79311016
asked Dec 12 '18 at 7:54
marmarmarmar
11
add a comment |
Using the fact that $y_1$=$t^{-1}$ then $y_2$= z $cdot$ $t^{-1}$
$y_2'= z'cdot t^{-1} - z cdot t^{-2}$
$y_2''= z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2 cdot z cdot t^{-3}$
Substituting into the original equation:
$t^2 (z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2cdot z cdot t^{-3})-2t(z cdot t^{-1})$
Which I then simplified into:
$tcdot z''-4cdot z'+2zcdot t^{-1} -2z = 3 t^2 -1$
I want to use the method of reduction of order to find the second solution, by setting $w=z'$, but there are too many orders of $z$ remaining in the equation. Not sure where I keep going wrong!
differential-equations reduction-of-order-ode
edited Dec 12 '18 at 8:04
postmortes
1,79311016
asked Dec 12 '18 at 7:54
marmarmarmar
11
This equation can not be solved using the known elementary functions
– Dr. Sonnhard Graubner
Dec 12 '18 at 8:12
1
There is a typo in the problem definition.
– Claude Leibovici
Dec 12 '18 at 8:14
1
@mamar Your equation hasn't solution and $y_1$ doesn't satisfy it as well.
– Nosrati
Dec 12 '18 at 8:22
add a comment |
Using the fact that $y_1$=$t^{-1}$ then $y_2$= z $cdot$ $t^{-1}$
$y_2'= z'cdot t^{-1} - z cdot t^{-2}$
$y_2''= z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2 cdot z cdot t^{-3}$
Substituting into the original equation:
$t^2 (z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2cdot z cdot t^{-3})-2t(z cdot t^{-1})$
Which I then simplified into:
$tcdot z''-4cdot z'+2zcdot t^{-1} -2z = 3 t^2 -1$
I want to use the method of reduction of order to find the second solution, by setting $w=z'$, but there are too many orders of $z$ remaining in the equation. Not sure where I keep going wrong!
differential-equations reduction-of-order-ode
edited Dec 12 '18 at 8:04
postmortes
1,79311016
asked Dec 12 '18 at 7:54
marmarmarmar
11
Using the fact that $y_1$=$t^{-1}$ then $y_2$= z $cdot$ $t^{-1}$
$y_2'= z'cdot t^{-1} - z cdot t^{-2}$
$y_2''= z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2 cdot z cdot t^{-3}$
Substituting into the original equation:
$t^2 (z''cdot t^{-1} - z'cdot t^{-2} - z'cdot t^{-2} + 2cdot z cdot t^{-3})-2t(z cdot t^{-1})$
Which I then simplified into:
$tcdot z''-4cdot z'+2zcdot t^{-1} -2z = 3 t^2 -1$
I want to use the method of reduction of order to find the second solution, by setting $w=z'$, but there are too many orders of $z$ remaining in the equation. Not sure where I keep going wrong!
differential-equations reduction-of-order-ode
differential-equations reduction-of-order-ode
edited Dec 12 '18 at 8:04
postmortes
1,79311016
asked Dec 12 '18 at 7:54
marmarmarmar
11
edited Dec 12 '18 at 8:04
postmortes
1,79311016
asked Dec 12 '18 at 7:54
marmarmarmar
11
edited Dec 12 '18 at 8:04
postmortes
1,79311016
edited Dec 12 '18 at 8:04
postmortes
1,79311016
edited Dec 12 '18 at 8:04
postmortes
1,79311016
1,79311016
asked Dec 12 '18 at 7:54
marmarmarmar
11
asked Dec 12 '18 at 7:54
marmarmarmar
11
asked Dec 12 '18 at 7:54
marmarmarmar
11
11
This equation can not be solved using the known elementary functions
– Dr. Sonnhard Graubner
Dec 12 '18 at 8:12
1
There is a typo in the problem definition.
– Claude Leibovici
Dec 12 '18 at 8:14
1
@mamar Your equation hasn't solution and $y_1$ doesn't satisfy it as well.
– Nosrati
Dec 12 '18 at 8:22
add a comment |
This equation can not be solved using the known elementary functions
– Dr. Sonnhard Graubner
Dec 12 '18 at 8:12
1
There is a typo in the problem definition.
– Claude Leibovici
Dec 12 '18 at 8:14
1
@mamar Your equation hasn't solution and $y_1$ doesn't satisfy it as well.
– Nosrati
Dec 12 '18 at 8:22
This equation can not be solved using the known elementary functions
– Dr. Sonnhard Graubner
Dec 12 '18 at 8:12
This equation can not be solved using the known elementary functions
– Dr. Sonnhard Graubner
Dec 12 '18 at 8:12
1
1
There is a typo in the problem definition.
– Claude Leibovici
Dec 12 '18 at 8:14
There is a typo in the problem definition.
– Claude Leibovici
Dec 12 '18 at 8:14
1
1
@mamar Your equation hasn't solution and $y_1$ doesn't satisfy it as well.
– Nosrati
Dec 12 '18 at 8:22
@mamar Your equation hasn't solution and $y_1$ doesn't satisfy it as well.
– Nosrati
Dec 12 '18 at 8:22
add a comment |
1 Answer
1
active
oldest
votes
If $y_1=t^{-1}$ is to be a solution of the homogeneous equation with minimal changes to the given equation, then the equation has to read
$$
t^2y''-2y=3t^2-1.
$$
Then the homogeneous equation is Euler-Cauchy with characteristic polynomial $0=m(m-1)-2=(m-2)(m+1)$. Thus the second basis solution is $y_2=t^2$.
Then a particular solution can be found in the form $y_0=Aln(t)t^2+B$.
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
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%2f3036380%2fi-need-some-help-finding-a-second-solution-to-t2-y-2ty-3t2-1-given-the-fac%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
If $y_1=t^{-1}$ is to be a solution of the homogeneous equation with minimal changes to the given equation, then the equation has to read
$$
t^2y''-2y=3t^2-1.
$$
Then the homogeneous equation is Euler-Cauchy with characteristic polynomial $0=m(m-1)-2=(m-2)(m+1)$. Thus the second basis solution is $y_2=t^2$.
Then a particular solution can be found in the form $y_0=Aln(t)t^2+B$.
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
add a comment |
If $y_1=t^{-1}$ is to be a solution of the homogeneous equation with minimal changes to the given equation, then the equation has to read
$$
t^2y''-2y=3t^2-1.
$$
Then the homogeneous equation is Euler-Cauchy with characteristic polynomial $0=m(m-1)-2=(m-2)(m+1)$. Thus the second basis solution is $y_2=t^2$.
Then a particular solution can be found in the form $y_0=Aln(t)t^2+B$.
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
add a comment |
If $y_1=t^{-1}$ is to be a solution of the homogeneous equation with minimal changes to the given equation, then the equation has to read
$$
t^2y''-2y=3t^2-1.
$$
Then the homogeneous equation is Euler-Cauchy with characteristic polynomial $0=m(m-1)-2=(m-2)(m+1)$. Thus the second basis solution is $y_2=t^2$.
Then a particular solution can be found in the form $y_0=Aln(t)t^2+B$.
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
If $y_1=t^{-1}$ is to be a solution of the homogeneous equation with minimal changes to the given equation, then the equation has to read
$$
t^2y''-2y=3t^2-1.
$$
Then the homogeneous equation is Euler-Cauchy with characteristic polynomial $0=m(m-1)-2=(m-2)(m+1)$. Thus the second basis solution is $y_2=t^2$.
Then a particular solution can be found in the form $y_0=Aln(t)t^2+B$.
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
answered Dec 12 '18 at 8:48
LutzLLutzL
56.5k42054
56.5k42054
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3036380%2fi-need-some-help-finding-a-second-solution-to-t2-y-2ty-3t2-1-given-the-fac%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
This equation can not be solved using the known elementary functions
– Dr. Sonnhard Graubner
Dec 12 '18 at 8:12
1
There is a typo in the problem definition.
– Claude Leibovici
Dec 12 '18 at 8:14
1
@mamar Your equation hasn't solution and $y_1$ doesn't satisfy it as well.
– Nosrati
Dec 12 '18 at 8:22