Applications of coupled systems of $;2times 2;$ linear differential equations
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I am providing maths help to some students studying just before University level in mathematics.
I am writing some practice questions for them on solving coupled first order linear equations and I was wondering if there are any 'real-life' applications of this.
I am having difficulty finding examples because the systems they need to solve are restricted to two dependent variables and one independent variable and need to be linear. (Most of the easy go-to options, e.g. SIR models, are not linear).
I have already used:
- predator-prey problems
- two-tanks mixing problems
Are there any others?
[For information, the equations they are required to solve should be of the form
$frac{dx}{dt} = ax + by + f(t), hspace{3mm} frac{dy}{dt} = cx + dy + g(t)$.]
ordinary-differential-equations soft-question applications
edited Dec 26 '18 at 19:03
user376343
3,9584829
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
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add a comment |
$begingroup$
I am providing maths help to some students studying just before University level in mathematics.
I am writing some practice questions for them on solving coupled first order linear equations and I was wondering if there are any 'real-life' applications of this.
I am having difficulty finding examples because the systems they need to solve are restricted to two dependent variables and one independent variable and need to be linear. (Most of the easy go-to options, e.g. SIR models, are not linear).
I have already used:
- predator-prey problems
- two-tanks mixing problems
Are there any others?
[For information, the equations they are required to solve should be of the form
$frac{dx}{dt} = ax + by + f(t), hspace{3mm} frac{dy}{dt} = cx + dy + g(t)$.]
ordinary-differential-equations soft-question applications
edited Dec 26 '18 at 19:03
user376343
3,9584829
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
$endgroup$
add a comment |
$begingroup$
I am providing maths help to some students studying just before University level in mathematics.
I am writing some practice questions for them on solving coupled first order linear equations and I was wondering if there are any 'real-life' applications of this.
I am having difficulty finding examples because the systems they need to solve are restricted to two dependent variables and one independent variable and need to be linear. (Most of the easy go-to options, e.g. SIR models, are not linear).
I have already used:
- predator-prey problems
- two-tanks mixing problems
Are there any others?
[For information, the equations they are required to solve should be of the form
$frac{dx}{dt} = ax + by + f(t), hspace{3mm} frac{dy}{dt} = cx + dy + g(t)$.]
ordinary-differential-equations soft-question applications
edited Dec 26 '18 at 19:03
user376343
3,9584829
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
$endgroup$
I am providing maths help to some students studying just before University level in mathematics.
I am writing some practice questions for them on solving coupled first order linear equations and I was wondering if there are any 'real-life' applications of this.
I am having difficulty finding examples because the systems they need to solve are restricted to two dependent variables and one independent variable and need to be linear. (Most of the easy go-to options, e.g. SIR models, are not linear).
I have already used:
- predator-prey problems
- two-tanks mixing problems
Are there any others?
[For information, the equations they are required to solve should be of the form
$frac{dx}{dt} = ax + by + f(t), hspace{3mm} frac{dy}{dt} = cx + dy + g(t)$.]
ordinary-differential-equations soft-question applications
ordinary-differential-equations soft-question applications
edited Dec 26 '18 at 19:03
user376343
3,9584829
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
edited Dec 26 '18 at 19:03
user376343
3,9584829
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
edited Dec 26 '18 at 19:03
user376343
3,9584829
edited Dec 26 '18 at 19:03
user376343
3,9584829
edited Dec 26 '18 at 19:03
user376343
3,9584829
3,9584829
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
asked Dec 26 '18 at 18:32
PhysicsMathsLovePhysicsMathsLove
1,210414
1,210414
add a comment |
add a comment |
1 Answer
1
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oldest
votes
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Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs. The heavy lines are strong springs. If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit. You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs. The heavy lines are strong springs. If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit. You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
add a comment |
$begingroup$
Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs. The heavy lines are strong springs. If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit. You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
add a comment |
$begingroup$
Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs. The heavy lines are strong springs. If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit. You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
$endgroup$
Coupled harmonic oscillators are a good example. The outer rectangles are walls, the inner ones are masses and the lines are springs. The heavy lines are strong springs. If you write it in terms of the positions of each mass, the coupling spring shifts the frequencies up and down a bit. You can do the linear algebra to show that the modes are symmetric and antisymmetric and that the correct basis is in those terms.
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
answered Dec 26 '18 at 18:41
Ross MillikanRoss Millikan
299k24200374
299k24200374
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
add a comment |
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
How would you set up first order differential equations for these particles, rather than second order?
$endgroup$
– David Quinn
Dec 26 '18 at 18:51
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
$begingroup$
You are right, I was thinking of linear but not first order. You can make it first order in four variables by defining the velocities as new variables.
$endgroup$
– Ross Millikan
Dec 27 '18 at 5:22
add a comment |
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