Mixing
Analytic or perturbative solution in any limits?
$begingroup$
Consider the system of 3 ordinary differential equations
$$dot{x}=v$$
$$dot{v}=a$$
$$dot{a}=-Aa+v^{2}-x$$
which can also be written as a single 3rd order ODE
$$dddot{x}=-Addot{x}+dot{x}^{2}-x$$
$A$ is an arbitrary constant and the dot means derivative with respect to time, i.e. $dot{x}=dx/dt,ddot{x}=d^{2}x/dt^{2}$, etc. This system can be thought as describing the time evolution of the position $x$, velocity $v$ and acceleration $a$ of a particle.
Are there any limits where we can solve analytically this system, i.e. find $x(t),v(t),a(t)$?
For example when $A=0$? A perturbative solution would also be good. Or maybe there is a way of reparametrizing time to make the system a known integrable one?
I know that the simpler system
$$dddot{x}=-Addot{x}iff dot{a}=-Aa$$
has the solution
$$a(t)=c_{1}e^{-At}$$ which means that
$$x(t)=frac{c_{1}}{A^{2}}e^{-At}+c_{2}t+c_{3}$$
integration ordinary-differential-equations power-series
asked Jan 5 at 20:29
user2175783user2175783
1876
$endgroup$
add a comment |
$begingroup$
Consider the system of 3 ordinary differential equations
$$dot{x}=v$$
$$dot{v}=a$$
$$dot{a}=-Aa+v^{2}-x$$
which can also be written as a single 3rd order ODE
$$dddot{x}=-Addot{x}+dot{x}^{2}-x$$
$A$ is an arbitrary constant and the dot means derivative with respect to time, i.e. $dot{x}=dx/dt,ddot{x}=d^{2}x/dt^{2}$, etc. This system can be thought as describing the time evolution of the position $x$, velocity $v$ and acceleration $a$ of a particle.
Are there any limits where we can solve analytically this system, i.e. find $x(t),v(t),a(t)$?
For example when $A=0$? A perturbative solution would also be good. Or maybe there is a way of reparametrizing time to make the system a known integrable one?
I know that the simpler system
$$dddot{x}=-Addot{x}iff dot{a}=-Aa$$
has the solution
$$a(t)=c_{1}e^{-At}$$ which means that
$$x(t)=frac{c_{1}}{A^{2}}e^{-At}+c_{2}t+c_{3}$$
integration ordinary-differential-equations power-series
asked Jan 5 at 20:29
user2175783user2175783
1876
$endgroup$
2
$begingroup$
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
$endgroup$
– JennyToy
Jan 5 at 23:55
add a comment |
$begingroup$
Consider the system of 3 ordinary differential equations
$$dot{x}=v$$
$$dot{v}=a$$
$$dot{a}=-Aa+v^{2}-x$$
which can also be written as a single 3rd order ODE
$$dddot{x}=-Addot{x}+dot{x}^{2}-x$$
$A$ is an arbitrary constant and the dot means derivative with respect to time, i.e. $dot{x}=dx/dt,ddot{x}=d^{2}x/dt^{2}$, etc. This system can be thought as describing the time evolution of the position $x$, velocity $v$ and acceleration $a$ of a particle.
Are there any limits where we can solve analytically this system, i.e. find $x(t),v(t),a(t)$?
For example when $A=0$? A perturbative solution would also be good. Or maybe there is a way of reparametrizing time to make the system a known integrable one?
I know that the simpler system
$$dddot{x}=-Addot{x}iff dot{a}=-Aa$$
has the solution
$$a(t)=c_{1}e^{-At}$$ which means that
$$x(t)=frac{c_{1}}{A^{2}}e^{-At}+c_{2}t+c_{3}$$
integration ordinary-differential-equations power-series
asked Jan 5 at 20:29
user2175783user2175783
1876
$endgroup$
Consider the system of 3 ordinary differential equations
$$dot{x}=v$$
$$dot{v}=a$$
$$dot{a}=-Aa+v^{2}-x$$
which can also be written as a single 3rd order ODE
$$dddot{x}=-Addot{x}+dot{x}^{2}-x$$
$A$ is an arbitrary constant and the dot means derivative with respect to time, i.e. $dot{x}=dx/dt,ddot{x}=d^{2}x/dt^{2}$, etc. This system can be thought as describing the time evolution of the position $x$, velocity $v$ and acceleration $a$ of a particle.
Are there any limits where we can solve analytically this system, i.e. find $x(t),v(t),a(t)$?
For example when $A=0$? A perturbative solution would also be good. Or maybe there is a way of reparametrizing time to make the system a known integrable one?
I know that the simpler system
$$dddot{x}=-Addot{x}iff dot{a}=-Aa$$
has the solution
$$a(t)=c_{1}e^{-At}$$ which means that
$$x(t)=frac{c_{1}}{A^{2}}e^{-At}+c_{2}t+c_{3}$$
integration ordinary-differential-equations power-series
integration ordinary-differential-equations power-series
asked Jan 5 at 20:29
user2175783user2175783
1876
asked Jan 5 at 20:29
user2175783user2175783
1876
edited Jan 6 at 5:24
user2175783
asked Jan 5 at 20:29
user2175783user2175783
1876
asked Jan 5 at 20:29
user2175783user2175783
1876
asked Jan 5 at 20:29
user2175783user2175783
1876
1876
2
$begingroup$
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
$endgroup$
– JennyToy
Jan 5 at 23:55
add a comment |
2
$begingroup$
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
$endgroup$
– JennyToy
Jan 5 at 23:55
2
2
$begingroup$
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
$endgroup$
– JennyToy
Jan 5 at 23:55
$begingroup$
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
$endgroup$
– JennyToy
Jan 5 at 23:55
add a comment |
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$begingroup$
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
$endgroup$
– JennyToy
Jan 5 at 23:55