Qualitatively describing classes of solutions to a system of linear differential equations
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I have a linear, autonomous, homogeneous system $$frac{doverrightarrow{w}}{dt}=Aoverrightarrow{w}$$ where $A$ is a $3 times 3$ matrix that represents a $120$ degree counterclockwise rotation about $begin{pmatrix} 0 \ 1 \ 0 end{pmatrix}$.
I need to find a general solution to the system but I'm curious what the different types of solutions to this system would look like. I'm pretty sure this can also depend on the initial values. Can anyone describe the classes of solutions in words?
differential-equations
asked Nov 28 at 1:09
Dominic Hicks
896
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I have a linear, autonomous, homogeneous system $$frac{doverrightarrow{w}}{dt}=Aoverrightarrow{w}$$ where $A$ is a $3 times 3$ matrix that represents a $120$ degree counterclockwise rotation about $begin{pmatrix} 0 \ 1 \ 0 end{pmatrix}$.
I need to find a general solution to the system but I'm curious what the different types of solutions to this system would look like. I'm pretty sure this can also depend on the initial values. Can anyone describe the classes of solutions in words?
differential-equations
asked Nov 28 at 1:09
Dominic Hicks
896
My guess without doing it out too much is that since $A^3=1$ you can diagonalize the system to three decoupled odes say $z_i'=omega^i z_i$ where $i=1,2,3$ and $omega^3=1$. Two of the solutions decay exponentially and oscilllate and one exponentially increases. Then the $w$ vector components are mixtures of these, two decaying and oscillating and one exponentially increasing.
– snulty
Nov 28 at 2:05
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have a linear, autonomous, homogeneous system $$frac{doverrightarrow{w}}{dt}=Aoverrightarrow{w}$$ where $A$ is a $3 times 3$ matrix that represents a $120$ degree counterclockwise rotation about $begin{pmatrix} 0 \ 1 \ 0 end{pmatrix}$.
I need to find a general solution to the system but I'm curious what the different types of solutions to this system would look like. I'm pretty sure this can also depend on the initial values. Can anyone describe the classes of solutions in words?
differential-equations
asked Nov 28 at 1:09
Dominic Hicks
896
I have a linear, autonomous, homogeneous system $$frac{doverrightarrow{w}}{dt}=Aoverrightarrow{w}$$ where $A$ is a $3 times 3$ matrix that represents a $120$ degree counterclockwise rotation about $begin{pmatrix} 0 \ 1 \ 0 end{pmatrix}$.
I need to find a general solution to the system but I'm curious what the different types of solutions to this system would look like. I'm pretty sure this can also depend on the initial values. Can anyone describe the classes of solutions in words?
differential-equations
differential-equations
asked Nov 28 at 1:09
Dominic Hicks
896
asked Nov 28 at 1:09
Dominic Hicks
896
edited Nov 28 at 17:13
asked Nov 28 at 1:09
Dominic Hicks
896
asked Nov 28 at 1:09
Dominic Hicks
896
asked Nov 28 at 1:09
Dominic Hicks
896
896
My guess without doing it out too much is that since $A^3=1$ you can diagonalize the system to three decoupled odes say $z_i'=omega^i z_i$ where $i=1,2,3$ and $omega^3=1$. Two of the solutions decay exponentially and oscilllate and one exponentially increases. Then the $w$ vector components are mixtures of these, two decaying and oscillating and one exponentially increasing.
– snulty
Nov 28 at 2:05
add a comment |
My guess without doing it out too much is that since $A^3=1$ you can diagonalize the system to three decoupled odes say $z_i'=omega^i z_i$ where $i=1,2,3$ and $omega^3=1$. Two of the solutions decay exponentially and oscilllate and one exponentially increases. Then the $w$ vector components are mixtures of these, two decaying and oscillating and one exponentially increasing.
– snulty
Nov 28 at 2:05
My guess without doing it out too much is that since $A^3=1$ you can diagonalize the system to three decoupled odes say $z_i'=omega^i z_i$ where $i=1,2,3$ and $omega^3=1$. Two of the solutions decay exponentially and oscilllate and one exponentially increases. Then the $w$ vector components are mixtures of these, two decaying and oscillating and one exponentially increasing.
– snulty
Nov 28 at 2:05
My guess without doing it out too much is that since $A^3=1$ you can diagonalize the system to three decoupled odes say $z_i'=omega^i z_i$ where $i=1,2,3$ and $omega^3=1$. Two of the solutions decay exponentially and oscilllate and one exponentially increases. Then the $w$ vector components are mixtures of these, two decaying and oscillating and one exponentially increasing.
– snulty
Nov 28 at 2:05
add a comment |
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My guess without doing it out too much is that since $A^3=1$ you can diagonalize the system to three decoupled odes say $z_i'=omega^i z_i$ where $i=1,2,3$ and $omega^3=1$. Two of the solutions decay exponentially and oscilllate and one exponentially increases. Then the $w$ vector components are mixtures of these, two decaying and oscillating and one exponentially increasing.
– snulty
Nov 28 at 2:05