Uniqueness and existence of this system, verifying my answer
$begingroup$
I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve.
1)Getting the Jacobian, I obtain
$$ J= begin{bmatrix}
0 & 1\
-1-2xy & 1-x^2-3y^2\
end{bmatrix} $$
From here I can say the Jacobain exists, and each of the entries are definitely continuous in the interval $$ x^2+y^2<4$$, so by Picard lindeloff we can say there are both existence and uniqueness of solutions in the interval.
2) $$ x(t)=sint, y(t)=cos(t)$$
Is this just as simple as getting the derivatives and subbing them into the equations to get 0=0 when you do? This seems over simplified but it does work and I know its easier than trying to solve for an explicit solution to verify the values, and it does work so why not?
3) I'm still working on part 3 and I'll edit it in when I figure it out.
ordinary-differential-equations jacobian nonlinear-analysis
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
$endgroup$
add a comment |
$begingroup$
I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve.
1)Getting the Jacobian, I obtain
$$ J= begin{bmatrix}
0 & 1\
-1-2xy & 1-x^2-3y^2\
end{bmatrix} $$
From here I can say the Jacobain exists, and each of the entries are definitely continuous in the interval $$ x^2+y^2<4$$, so by Picard lindeloff we can say there are both existence and uniqueness of solutions in the interval.
2) $$ x(t)=sint, y(t)=cos(t)$$
Is this just as simple as getting the derivatives and subbing them into the equations to get 0=0 when you do? This seems over simplified but it does work and I know its easier than trying to solve for an explicit solution to verify the values, and it does work so why not?
3) I'm still working on part 3 and I'll edit it in when I figure it out.
ordinary-differential-equations jacobian nonlinear-analysis
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
$endgroup$
$begingroup$
Well since I could calculate the Jacobian, it definitely exists and each entry ia a countinuous function in $R^2$, so also a continuos function in the disk.
$endgroup$
– Robbie Meaney
Dec 10 '18 at 12:35
add a comment |
$begingroup$
I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve.
1)Getting the Jacobian, I obtain
$$ J= begin{bmatrix}
0 & 1\
-1-2xy & 1-x^2-3y^2\
end{bmatrix} $$
From here I can say the Jacobain exists, and each of the entries are definitely continuous in the interval $$ x^2+y^2<4$$, so by Picard lindeloff we can say there are both existence and uniqueness of solutions in the interval.
2) $$ x(t)=sint, y(t)=cos(t)$$
Is this just as simple as getting the derivatives and subbing them into the equations to get 0=0 when you do? This seems over simplified but it does work and I know its easier than trying to solve for an explicit solution to verify the values, and it does work so why not?
3) I'm still working on part 3 and I'll edit it in when I figure it out.
ordinary-differential-equations jacobian nonlinear-analysis
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
$endgroup$
I have an exam in this tomorrow, and I want to make sure my answers are correct, and if not what I can do to improve.
1)Getting the Jacobian, I obtain
$$ J= begin{bmatrix}
0 & 1\
-1-2xy & 1-x^2-3y^2\
end{bmatrix} $$
From here I can say the Jacobain exists, and each of the entries are definitely continuous in the interval $$ x^2+y^2<4$$, so by Picard lindeloff we can say there are both existence and uniqueness of solutions in the interval.
2) $$ x(t)=sint, y(t)=cos(t)$$
Is this just as simple as getting the derivatives and subbing them into the equations to get 0=0 when you do? This seems over simplified but it does work and I know its easier than trying to solve for an explicit solution to verify the values, and it does work so why not?
3) I'm still working on part 3 and I'll edit it in when I figure it out.
ordinary-differential-equations jacobian nonlinear-analysis
ordinary-differential-equations jacobian nonlinear-analysis
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
asked Dec 10 '18 at 12:21
Robbie MeaneyRobbie Meaney
749
749
$begingroup$
Well since I could calculate the Jacobian, it definitely exists and each entry ia a countinuous function in $R^2$, so also a continuos function in the disk.
$endgroup$
– Robbie Meaney
Dec 10 '18 at 12:35
add a comment |
$begingroup$
Well since I could calculate the Jacobian, it definitely exists and each entry ia a countinuous function in $R^2$, so also a continuos function in the disk.
$endgroup$
– Robbie Meaney
Dec 10 '18 at 12:35
$begingroup$
Well since I could calculate the Jacobian, it definitely exists and each entry ia a countinuous function in $R^2$, so also a continuos function in the disk.
$endgroup$
– Robbie Meaney
Dec 10 '18 at 12:35
$begingroup$
Well since I could calculate the Jacobian, it definitely exists and each entry ia a countinuous function in $R^2$, so also a continuos function in the disk.
$endgroup$
– Robbie Meaney
Dec 10 '18 at 12:35
add a comment |
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$begingroup$
Well since I could calculate the Jacobian, it definitely exists and each entry ia a countinuous function in $R^2$, so also a continuos function in the disk.
$endgroup$
– Robbie Meaney
Dec 10 '18 at 12:35