Timeline for Can we solve this ODE?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 15, 2021 at 22:14 | vote | accept | Kung Yao | ||
| Jan 15, 2021 at 19:56 | answer | added | მამუკა ჯიბლაძე | timeline score: 3 | |
| Jan 12, 2021 at 18:12 | comment | added | Will Jagy | found an undergraduate project on Floquet, fse.studenttheses.ub.rug.nl/17640/1/bMATH_2018_FolkersE.pdf Not sure yet if we can explicitly calculate the monodromy matrix if we cannot explicitly write down a fundamental matrix | |
| Jan 12, 2021 at 2:29 | comment | added | Will Jagy | Look up Floquet theory for periodic coefficients en.wikipedia.org/wiki/Floquet_theory | |
| Jan 11, 2021 at 19:35 | comment | added | Christian Remling | Variable coefficient linear ODEs as a general rule don't have explicit solutions as soon as the order exceeds $1$. | |
| Jan 11, 2021 at 19:23 | comment | added | Pietro Majer | In general, $\big(e^{B(x)}\big)'$ is not $e^{B(x)}B'(x)$ nor $B'(x)e^{B(x)}$, unless $B$ and $B'$ commute | |
| Jan 11, 2021 at 19:21 | comment | added | Will Jagy | @მამუკაჯიბლაძე it works when the coefficient matrix commutes with its derivative | |
| Jan 11, 2021 at 19:18 | comment | added | მამუკა ჯიბლაძე | Yes, you are right. By some reason I expected that $y$ and the antiderivative would commute, but now I cannot actually find any reason to expect it. | |
| Jan 11, 2021 at 19:01 | comment | added | Kung Yao | @მამუკაჯიბლაძე mhmm, but so what you did is that you computed the antiderivative $B(x)$ of the matrix, let's call it $A(x)$. But this way wouldn't we have. So $y(x) = exp(B(x))$ but then $y'(x)= y(x)A(x)$ by the chain rule. So why is this the same as $A(x)y(x)$? | |
| Jan 11, 2021 at 18:55 | comment | added | მამუკა ჯიბლაძე | This is completely standard except for matrices: $\frac{dy}y=(\text{your matrix})dx$ gives$$y=\exp\left(\left(\begin{smallmatrix}0&x+\sin(x+\frac\pi4)\\x-\sin(x-\frac\pi4)&0\end{smallmatrix}\right)+\text{any constant matrix}\right)$$ | |
| Jan 11, 2021 at 18:50 | comment | added | Mateusz Kwaśnicki | Mathematica 10.0 does not seem to give any closed-form solution. | |
| Jan 11, 2021 at 18:42 | history | asked | Kung Yao | CC BY-SA 4.0 |