Fundamental matrix of ODE system $dot{x} = (At+B)x$
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Suppose I have a system of ODEs $dot{x} = (At+B)x$, where $x(t)$ is a, say, $n times 1$ vector, and $A$ and $B$ are constant $n times n$ matrices. What is the fundamental matrix of this system?
I tried $e^{frac{At^2}{2}}e^{Bt}$, trying to emulate the one-dimensional solution, but then A and B don't necessarily commute for it to work. Do I need the Lie bracket $[A,B] = AB - BA$ somewhere? Do I need something like power series?
ordinary-differential-equations non-linear-dynamics
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
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add a comment |
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Suppose I have a system of ODEs $dot{x} = (At+B)x$, where $x(t)$ is a, say, $n times 1$ vector, and $A$ and $B$ are constant $n times n$ matrices. What is the fundamental matrix of this system?
I tried $e^{frac{At^2}{2}}e^{Bt}$, trying to emulate the one-dimensional solution, but then A and B don't necessarily commute for it to work. Do I need the Lie bracket $[A,B] = AB - BA$ somewhere? Do I need something like power series?
ordinary-differential-equations non-linear-dynamics
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
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3
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You can't expect an answer in terms of elementary functions, since (for example) if $At+B = begin{pmatrix} 0 & 1 \ t & 0 end{pmatrix}$ the system is equivalent to the Airy equation $ddot x_1(t) = t , x_1(t)$, with Airy functions as solutions. But maybe you could try looking for a solution in the form of a power series?
$endgroup$
– Hans Lundmark
Dec 21 '18 at 14:46
add a comment |
$begingroup$
Suppose I have a system of ODEs $dot{x} = (At+B)x$, where $x(t)$ is a, say, $n times 1$ vector, and $A$ and $B$ are constant $n times n$ matrices. What is the fundamental matrix of this system?
I tried $e^{frac{At^2}{2}}e^{Bt}$, trying to emulate the one-dimensional solution, but then A and B don't necessarily commute for it to work. Do I need the Lie bracket $[A,B] = AB - BA$ somewhere? Do I need something like power series?
ordinary-differential-equations non-linear-dynamics
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
$endgroup$
Suppose I have a system of ODEs $dot{x} = (At+B)x$, where $x(t)$ is a, say, $n times 1$ vector, and $A$ and $B$ are constant $n times n$ matrices. What is the fundamental matrix of this system?
I tried $e^{frac{At^2}{2}}e^{Bt}$, trying to emulate the one-dimensional solution, but then A and B don't necessarily commute for it to work. Do I need the Lie bracket $[A,B] = AB - BA$ somewhere? Do I need something like power series?
ordinary-differential-equations non-linear-dynamics
ordinary-differential-equations non-linear-dynamics
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
asked Dec 21 '18 at 14:02
AspiringMathematicianAspiringMathematician
1,781621
1,781621
3
$begingroup$
You can't expect an answer in terms of elementary functions, since (for example) if $At+B = begin{pmatrix} 0 & 1 \ t & 0 end{pmatrix}$ the system is equivalent to the Airy equation $ddot x_1(t) = t , x_1(t)$, with Airy functions as solutions. But maybe you could try looking for a solution in the form of a power series?
$endgroup$
– Hans Lundmark
Dec 21 '18 at 14:46
add a comment |
3
$begingroup$
You can't expect an answer in terms of elementary functions, since (for example) if $At+B = begin{pmatrix} 0 & 1 \ t & 0 end{pmatrix}$ the system is equivalent to the Airy equation $ddot x_1(t) = t , x_1(t)$, with Airy functions as solutions. But maybe you could try looking for a solution in the form of a power series?
$endgroup$
– Hans Lundmark
Dec 21 '18 at 14:46
3
3
$begingroup$
You can't expect an answer in terms of elementary functions, since (for example) if $At+B = begin{pmatrix} 0 & 1 \ t & 0 end{pmatrix}$ the system is equivalent to the Airy equation $ddot x_1(t) = t , x_1(t)$, with Airy functions as solutions. But maybe you could try looking for a solution in the form of a power series?
$endgroup$
– Hans Lundmark
Dec 21 '18 at 14:46
$begingroup$
You can't expect an answer in terms of elementary functions, since (for example) if $At+B = begin{pmatrix} 0 & 1 \ t & 0 end{pmatrix}$ the system is equivalent to the Airy equation $ddot x_1(t) = t , x_1(t)$, with Airy functions as solutions. But maybe you could try looking for a solution in the form of a power series?
$endgroup$
– Hans Lundmark
Dec 21 '18 at 14:46
add a comment |
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$begingroup$
You can't expect an answer in terms of elementary functions, since (for example) if $At+B = begin{pmatrix} 0 & 1 \ t & 0 end{pmatrix}$ the system is equivalent to the Airy equation $ddot x_1(t) = t , x_1(t)$, with Airy functions as solutions. But maybe you could try looking for a solution in the form of a power series?
$endgroup$
– Hans Lundmark
Dec 21 '18 at 14:46