Solving a linear system of differential equations
Given that $v_1 = begin{bmatrix}1&1end{bmatrix}$ and $v_2 = begin{bmatrix}2 &1end{bmatrix}$ are eigenvectors of the matrix
$$
begin{bmatrix}-1&-2\1&-4end{bmatrix}
$$
which is a $2times 2$ matrix.
Find the solution to the linear system of differential equations
begin{align*}
x' &= -x - 2y\
y' &= x - 4y
end{align*}
satisfying the initial conditions $x(0)=7$ and $y(0)=5$.
So I already found the eigenvalues, $-3$ and $-2$ and I know that you need to plug the eigenvalues into the matrix you get from doing $det(It - A)$ but I'm not sure where to go from there in terms of making it into an equation?
linear-algebra differential-equations eigenvalues-eigenvectors
asked Dec 8 at 20:56
S. Snake
485
add a comment |
Given that $v_1 = begin{bmatrix}1&1end{bmatrix}$ and $v_2 = begin{bmatrix}2 &1end{bmatrix}$ are eigenvectors of the matrix
$$
begin{bmatrix}-1&-2\1&-4end{bmatrix}
$$
which is a $2times 2$ matrix.
Find the solution to the linear system of differential equations
begin{align*}
x' &= -x - 2y\
y' &= x - 4y
end{align*}
satisfying the initial conditions $x(0)=7$ and $y(0)=5$.
So I already found the eigenvalues, $-3$ and $-2$ and I know that you need to plug the eigenvalues into the matrix you get from doing $det(It - A)$ but I'm not sure where to go from there in terms of making it into an equation?
linear-algebra differential-equations eigenvalues-eigenvectors
asked Dec 8 at 20:56
S. Snake
485
add a comment |
Given that $v_1 = begin{bmatrix}1&1end{bmatrix}$ and $v_2 = begin{bmatrix}2 &1end{bmatrix}$ are eigenvectors of the matrix
$$
begin{bmatrix}-1&-2\1&-4end{bmatrix}
$$
which is a $2times 2$ matrix.
Find the solution to the linear system of differential equations
begin{align*}
x' &= -x - 2y\
y' &= x - 4y
end{align*}
satisfying the initial conditions $x(0)=7$ and $y(0)=5$.
So I already found the eigenvalues, $-3$ and $-2$ and I know that you need to plug the eigenvalues into the matrix you get from doing $det(It - A)$ but I'm not sure where to go from there in terms of making it into an equation?
linear-algebra differential-equations eigenvalues-eigenvectors
asked Dec 8 at 20:56
S. Snake
485
Given that $v_1 = begin{bmatrix}1&1end{bmatrix}$ and $v_2 = begin{bmatrix}2 &1end{bmatrix}$ are eigenvectors of the matrix
$$
begin{bmatrix}-1&-2\1&-4end{bmatrix}
$$
which is a $2times 2$ matrix.
Find the solution to the linear system of differential equations
begin{align*}
x' &= -x - 2y\
y' &= x - 4y
end{align*}
satisfying the initial conditions $x(0)=7$ and $y(0)=5$.
So I already found the eigenvalues, $-3$ and $-2$ and I know that you need to plug the eigenvalues into the matrix you get from doing $det(It - A)$ but I'm not sure where to go from there in terms of making it into an equation?
linear-algebra differential-equations eigenvalues-eigenvectors
linear-algebra differential-equations eigenvalues-eigenvectors
asked Dec 8 at 20:56
S. Snake
485
asked Dec 8 at 20:56
S. Snake
485
edited Dec 8 at 23:19
asked Dec 8 at 20:56
S. Snake
485
asked Dec 8 at 20:56
S. Snake
485
asked Dec 8 at 20:56
S. Snake
485
485
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
We can write the solution to the system as
$$X(t) = begin{bmatrix} x(t) \ y(t)end{bmatrix} = c_1 e^{lambda_1 t} v_1 + c_2 e^{lambda_2 t} v_2$$
From the given information, we have
$$X(t) = c_1 e^{-3 t}begin{bmatrix} 1 \ 1 end{bmatrix} + c_2 e^{-2 t}begin{bmatrix} 2 \ 1 end{bmatrix}$$
Now, use the initial conditions to solve for $c_1$ and $c_2$. You can see examples here.
answered Dec 9 at 0:18
Moo
5,53131020
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%2f3031628%2fsolving-a-linear-system-of-differential-equations%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
We can write the solution to the system as
$$X(t) = begin{bmatrix} x(t) \ y(t)end{bmatrix} = c_1 e^{lambda_1 t} v_1 + c_2 e^{lambda_2 t} v_2$$
From the given information, we have
$$X(t) = c_1 e^{-3 t}begin{bmatrix} 1 \ 1 end{bmatrix} + c_2 e^{-2 t}begin{bmatrix} 2 \ 1 end{bmatrix}$$
Now, use the initial conditions to solve for $c_1$ and $c_2$. You can see examples here.
answered Dec 9 at 0:18
Moo
5,53131020
add a comment |
We can write the solution to the system as
$$X(t) = begin{bmatrix} x(t) \ y(t)end{bmatrix} = c_1 e^{lambda_1 t} v_1 + c_2 e^{lambda_2 t} v_2$$
From the given information, we have
$$X(t) = c_1 e^{-3 t}begin{bmatrix} 1 \ 1 end{bmatrix} + c_2 e^{-2 t}begin{bmatrix} 2 \ 1 end{bmatrix}$$
Now, use the initial conditions to solve for $c_1$ and $c_2$. You can see examples here.
answered Dec 9 at 0:18
Moo
5,53131020
add a comment |
We can write the solution to the system as
$$X(t) = begin{bmatrix} x(t) \ y(t)end{bmatrix} = c_1 e^{lambda_1 t} v_1 + c_2 e^{lambda_2 t} v_2$$
From the given information, we have
$$X(t) = c_1 e^{-3 t}begin{bmatrix} 1 \ 1 end{bmatrix} + c_2 e^{-2 t}begin{bmatrix} 2 \ 1 end{bmatrix}$$
Now, use the initial conditions to solve for $c_1$ and $c_2$. You can see examples here.
answered Dec 9 at 0:18
Moo
5,53131020
We can write the solution to the system as
$$X(t) = begin{bmatrix} x(t) \ y(t)end{bmatrix} = c_1 e^{lambda_1 t} v_1 + c_2 e^{lambda_2 t} v_2$$
From the given information, we have
$$X(t) = c_1 e^{-3 t}begin{bmatrix} 1 \ 1 end{bmatrix} + c_2 e^{-2 t}begin{bmatrix} 2 \ 1 end{bmatrix}$$
Now, use the initial conditions to solve for $c_1$ and $c_2$. You can see examples here.
answered Dec 9 at 0:18
Moo
5,53131020
edited Dec 9 at 17:38
answered Dec 9 at 0:18
Moo
5,53131020
answered Dec 9 at 0:18
Moo
5,53131020
answered Dec 9 at 0:18
Moo
5,53131020
5,53131020
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%2f3031628%2fsolving-a-linear-system-of-differential-equations%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
