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Help with first order non-linear differential equation
Unusual 2nd order inhomogeneous equation..Help on second order linear equationsFinding the index of a linear vector field at the originSolving a first order linear ODE and determining the behavior of its solutionsFirst order non-linear ordinary differential equationInequality with differential equations solutionsRoot-finding with following set of equationsexistence of solution to set of non-linear equationsAutonomous system of 2 non-linear differential equationsNon-linear first order ODE with auxiliary variable
$begingroup$
I've been trying to solve this one for a while, but I still can't make it. Here's the problem.
I have a $f_1(t;rho,nu)$ that for $ttoinfty$ and for $rho>nu$ goes as $f_1sim t^nu/rho.$
Saying that, the problem is with a second function $f_2(t, rho, nu)$ which obeys the equation:
$fracdf_2(t)dt=fracnu f_2(t) + (nu+1)f_1(t)rho t + (nu+1)[f_1(t)+f_2(t)]$
with $rho,nuinmathbbN^+$ and $f_2(t)>0$ (and increasing).
I am interested in the long term behavior as for $f_1$. I know (from simulations) that for $ttoinfty$ it behaves similarly to $f_1$, but with a different exponent. I've tried to plug in the $f_1$ written above (assuming $rho>nu)$ and also to force $f_2(t)sim ct^alpha$ as a solution, ultimately looking for an expression for $alpha$ as a function of the other parameters, but I only arrived to contradict myself.
I've also tried on Mathematica with initial conditions $f_2(0)=0$ but without succeeding.
Any tips?
Thanks in advance!!
ordinary-differential-equations closed-form nonlinear-system non-linear-dynamics
asked Mar 29 at 15:26
MeffoMeffo
11
$endgroup$
add a comment |
$begingroup$
I've been trying to solve this one for a while, but I still can't make it. Here's the problem.
I have a $f_1(t;rho,nu)$ that for $ttoinfty$ and for $rho>nu$ goes as $f_1sim t^nu/rho.$
Saying that, the problem is with a second function $f_2(t, rho, nu)$ which obeys the equation:
$fracdf_2(t)dt=fracnu f_2(t) + (nu+1)f_1(t)rho t + (nu+1)[f_1(t)+f_2(t)]$
with $rho,nuinmathbbN^+$ and $f_2(t)>0$ (and increasing).
I am interested in the long term behavior as for $f_1$. I know (from simulations) that for $ttoinfty$ it behaves similarly to $f_1$, but with a different exponent. I've tried to plug in the $f_1$ written above (assuming $rho>nu)$ and also to force $f_2(t)sim ct^alpha$ as a solution, ultimately looking for an expression for $alpha$ as a function of the other parameters, but I only arrived to contradict myself.
I've also tried on Mathematica with initial conditions $f_2(0)=0$ but without succeeding.
Any tips?
Thanks in advance!!
ordinary-differential-equations closed-form nonlinear-system non-linear-dynamics
asked Mar 29 at 15:26
MeffoMeffo
11
$endgroup$
1
$begingroup$
If this is a differential equation, you don't want $t in mathbb N$.
$endgroup$
– Robert Israel
Mar 29 at 15:54
$begingroup$
OK thanks Robert. I put it there because in the original process it's defined in discrete times, but here I totally agree that it doesn't make sense ;)
$endgroup$
– Meffo
Mar 30 at 17:58
add a comment |
$begingroup$
I've been trying to solve this one for a while, but I still can't make it. Here's the problem.
I have a $f_1(t;rho,nu)$ that for $ttoinfty$ and for $rho>nu$ goes as $f_1sim t^nu/rho.$
Saying that, the problem is with a second function $f_2(t, rho, nu)$ which obeys the equation:
$fracdf_2(t)dt=fracnu f_2(t) + (nu+1)f_1(t)rho t + (nu+1)[f_1(t)+f_2(t)]$
with $rho,nuinmathbbN^+$ and $f_2(t)>0$ (and increasing).
I am interested in the long term behavior as for $f_1$. I know (from simulations) that for $ttoinfty$ it behaves similarly to $f_1$, but with a different exponent. I've tried to plug in the $f_1$ written above (assuming $rho>nu)$ and also to force $f_2(t)sim ct^alpha$ as a solution, ultimately looking for an expression for $alpha$ as a function of the other parameters, but I only arrived to contradict myself.
I've also tried on Mathematica with initial conditions $f_2(0)=0$ but without succeeding.
Any tips?
Thanks in advance!!
ordinary-differential-equations closed-form nonlinear-system non-linear-dynamics
asked Mar 29 at 15:26
MeffoMeffo
11
$endgroup$
I've been trying to solve this one for a while, but I still can't make it. Here's the problem.
I have a $f_1(t;rho,nu)$ that for $ttoinfty$ and for $rho>nu$ goes as $f_1sim t^nu/rho.$
Saying that, the problem is with a second function $f_2(t, rho, nu)$ which obeys the equation:
$fracdf_2(t)dt=fracnu f_2(t) + (nu+1)f_1(t)rho t + (nu+1)[f_1(t)+f_2(t)]$
with $rho,nuinmathbbN^+$ and $f_2(t)>0$ (and increasing).
I am interested in the long term behavior as for $f_1$. I know (from simulations) that for $ttoinfty$ it behaves similarly to $f_1$, but with a different exponent. I've tried to plug in the $f_1$ written above (assuming $rho>nu)$ and also to force $f_2(t)sim ct^alpha$ as a solution, ultimately looking for an expression for $alpha$ as a function of the other parameters, but I only arrived to contradict myself.
I've also tried on Mathematica with initial conditions $f_2(0)=0$ but without succeeding.
Any tips?
Thanks in advance!!
ordinary-differential-equations closed-form nonlinear-system non-linear-dynamics
ordinary-differential-equations closed-form nonlinear-system non-linear-dynamics
asked Mar 29 at 15:26
MeffoMeffo
11
asked Mar 29 at 15:26
MeffoMeffo
11
edited Mar 30 at 18:01
Meffo
asked Mar 29 at 15:26
MeffoMeffo
11
asked Mar 29 at 15:26
MeffoMeffo
11
asked Mar 29 at 15:26
MeffoMeffo
11
11
1
$begingroup$
If this is a differential equation, you don't want $t in mathbb N$.
$endgroup$
– Robert Israel
Mar 29 at 15:54
$begingroup$
OK thanks Robert. I put it there because in the original process it's defined in discrete times, but here I totally agree that it doesn't make sense ;)
$endgroup$
– Meffo
Mar 30 at 17:58
add a comment |
1
$begingroup$
If this is a differential equation, you don't want $t in mathbb N$.
$endgroup$
– Robert Israel
Mar 29 at 15:54
$begingroup$
OK thanks Robert. I put it there because in the original process it's defined in discrete times, but here I totally agree that it doesn't make sense ;)
$endgroup$
– Meffo
Mar 30 at 17:58
1
1
$begingroup$
If this is a differential equation, you don't want $t in mathbb N$.
$endgroup$
– Robert Israel
Mar 29 at 15:54
$begingroup$
If this is a differential equation, you don't want $t in mathbb N$.
$endgroup$
– Robert Israel
Mar 29 at 15:54
$begingroup$
OK thanks Robert. I put it there because in the original process it's defined in discrete times, but here I totally agree that it doesn't make sense ;)
$endgroup$
– Meffo
Mar 30 at 17:58
$begingroup$
OK thanks Robert. I put it there because in the original process it's defined in discrete times, but here I totally agree that it doesn't make sense ;)
$endgroup$
– Meffo
Mar 30 at 17:58
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It seems to me that it might have the same exponent, but logarithmic terms. Rather than try to solve the differential equation for $f_2(t)$, I tried plugging in a form for $f_2(t)$ and solving for $f_1(t)$. I found that
$$ f_2(t) = fracnu + 1rho log(t); t^nu/rho $$
is a solution with
$$ f_1(t) = frac 1rho left( t^frac nu+
rhorhorho^3+ln left( t right) t^2,frac nu
rho left( nu+1 right) ^2 left( ln left( t right) nu+
rho right) right) left( - left( nu+1 right) left( ln
left( t right) nu+rho right) t^frac nurho+rho^2
t right) ^-1 sim t^nu/rho
$$
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
It seems to me that it might have the same exponent, but logarithmic terms. Rather than try to solve the differential equation for $f_2(t)$, I tried plugging in a form for $f_2(t)$ and solving for $f_1(t)$. I found that
$$ f_2(t) = fracnu + 1rho log(t); t^nu/rho $$
is a solution with
$$ f_1(t) = frac 1rho left( t^frac nu+
rhorhorho^3+ln left( t right) t^2,frac nu
rho left( nu+1 right) ^2 left( ln left( t right) nu+
rho right) right) left( - left( nu+1 right) left( ln
left( t right) nu+rho right) t^frac nurho+rho^2
t right) ^-1 sim t^nu/rho
$$
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
add a comment |
$begingroup$
It seems to me that it might have the same exponent, but logarithmic terms. Rather than try to solve the differential equation for $f_2(t)$, I tried plugging in a form for $f_2(t)$ and solving for $f_1(t)$. I found that
$$ f_2(t) = fracnu + 1rho log(t); t^nu/rho $$
is a solution with
$$ f_1(t) = frac 1rho left( t^frac nu+
rhorhorho^3+ln left( t right) t^2,frac nu
rho left( nu+1 right) ^2 left( ln left( t right) nu+
rho right) right) left( - left( nu+1 right) left( ln
left( t right) nu+rho right) t^frac nurho+rho^2
t right) ^-1 sim t^nu/rho
$$
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
add a comment |
$begingroup$
It seems to me that it might have the same exponent, but logarithmic terms. Rather than try to solve the differential equation for $f_2(t)$, I tried plugging in a form for $f_2(t)$ and solving for $f_1(t)$. I found that
$$ f_2(t) = fracnu + 1rho log(t); t^nu/rho $$
is a solution with
$$ f_1(t) = frac 1rho left( t^frac nu+
rhorhorho^3+ln left( t right) t^2,frac nu
rho left( nu+1 right) ^2 left( ln left( t right) nu+
rho right) right) left( - left( nu+1 right) left( ln
left( t right) nu+rho right) t^frac nurho+rho^2
t right) ^-1 sim t^nu/rho
$$
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
$endgroup$
It seems to me that it might have the same exponent, but logarithmic terms. Rather than try to solve the differential equation for $f_2(t)$, I tried plugging in a form for $f_2(t)$ and solving for $f_1(t)$. I found that
$$ f_2(t) = fracnu + 1rho log(t); t^nu/rho $$
is a solution with
$$ f_1(t) = frac 1rho left( t^frac nu+
rhorhorho^3+ln left( t right) t^2,frac nu
rho left( nu+1 right) ^2 left( ln left( t right) nu+
rho right) right) left( - left( nu+1 right) left( ln
left( t right) nu+rho right) t^frac nurho+rho^2
t right) ^-1 sim t^nu/rho
$$
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
answered Mar 31 at 16:09
Robert IsraelRobert Israel
330k23219473
330k23219473
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
add a comment |
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
Thanks a lot robert for the hint! So, just to be sure, you plugged a $f_2(t)=ct^alpha$ into the diff. equation and the second Eq. you wrote is the solution you got? I'm missing the bit where you derive then the $f_2(t)$
$endgroup$
– Meffo
Apr 2 at 10:55
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
$begingroup$
No, I plugged in $f_2(t) = c log(t) t^nu/rho$ and solved for $f_1(t)$, then found what $c$ was needed for $f_1(t) sim t^nu/rho$.
$endgroup$
– Robert Israel
Apr 2 at 12:00
add a comment |
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$begingroup$
If this is a differential equation, you don't want $t in mathbb N$.
$endgroup$
– Robert Israel
Mar 29 at 15:54
$begingroup$
OK thanks Robert. I put it there because in the original process it's defined in discrete times, but here I totally agree that it doesn't make sense ;)
$endgroup$
– Meffo
Mar 30 at 17:58