Analytic or perturbative solution in any limits?
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 differential-equations power-series
asked 2 days ago
user2175783user2175783
1876
add a comment |
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 differential-equations power-series
asked 2 days ago
user2175783user2175783
1876
2
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
– JennyToy
2 days ago
add a comment |
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 differential-equations power-series
asked 2 days ago
user2175783user2175783
1876
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 differential-equations power-series
integration differential-equations power-series
asked 2 days ago
user2175783user2175783
1876
asked 2 days ago
user2175783user2175783
1876
edited 2 days ago
user2175783
asked 2 days ago
user2175783user2175783
1876
asked 2 days ago
user2175783user2175783
1876
asked 2 days ago
user2175783user2175783
1876
1876
2
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
– JennyToy
2 days ago
add a comment |
2
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
– JennyToy
2 days ago
2
2
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
– JennyToy
2 days ago
The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
– JennyToy
2 days ago
add a comment |
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The linear equation $dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}left(c_{2}cosfrac{sqrt{3}}{2}t+c_{3}sinfrac{sqrt{3}}{2}tright)$$
– JennyToy
2 days ago