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Autonomous ODE

by williamrand1
Tags: autonomous
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williamrand1
#1
Jan23-13, 02:35 PM
P: 21
Hi everyone,

Im looking for an autonomous first order ode that has the following properties.

For dependent variable x:

x(t=∞)=0

x(t=-∞)=0

and the function x(t) has one maximum.

Any help would be great.

Rgds...
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pasmith
#2
Jan23-13, 04:30 PM
HW Helper
Thanks
P: 1,001
This is impossible.

Suppose the maximum is at [itex]t = t_0[/itex]. Then there exist [itex]t_1 < t_0 < t_2[/itex] such that [itex]x(t_1) = x(t_2)[/itex], but [itex]\dot x(t_1) = -\dot x(t_2)[/itex]. There is no way to express that requirement in an autonomous first order ODE.

You are going to need a second-order autonomous ODE, as should be obvious from the fact that you want to satisfy two boundary conditions.
JJacquelin
#3
Jan24-13, 01:06 AM
P: 759
What do you think of y' = - y^(3/2) ?

williamrand1
#4
Jan24-13, 11:16 AM
P: 21
Autonomous ODE

Quote Quote by pasmith View Post
This is impossible.

Suppose the maximum is at [itex]t = t_0[/itex]. Then there exist [itex]t_1 < t_0 < t_2[/itex] such that [itex]x(t_1) = x(t_2)[/itex], but [itex]\dot x(t_1) = -\dot x(t_2)[/itex]. There is no way to express that requirement in an autonomous first order ODE.

You are going to need a second-order autonomous ODE, as should be obvious from the fact that you want to satisfy two boundary conditions.
Thanks pasmith

Could you explain why it is not possible?
williamrand1
#5
Jan24-13, 11:18 AM
P: 21
Quote Quote by JJacquelin View Post
What do you think of y' = - y^(3/2) ?
Thanks JJ

Is there an exact solution to this?
JJacquelin
#6
Jan24-13, 02:54 PM
P: 759
dy/dx = -y^(3/2)
dx = - dy/y^(3/2)
x = (2 / y^(1/2)) +C
y^(1/2) = 2/(x-C)
y = 4/(x-C)
Mute
#7
Jan24-13, 10:40 PM
HW Helper
P: 1,391
Quote Quote by JJacquelin View Post
dy/dx = -y^(3/2)
dx = - dy/y^(3/2)
x = (2 / y^(1/2)) +C
y^(1/2) = 2/(x-C)
y = 4/(x-C)
That has a divergence, not a maximum, though! I'm not sure that's what williamrand1 is looking for.

williamrand1, what about trying to take a function that you know has the properties you desire, differentiate it, and then see if you can rewrite the derivative in terms of x(t), with no explicit time dependence?
JJacquelin
#8
Jan25-13, 12:38 AM
P: 759
Hi williamrand1 !

Then, what about this one :
y' = -2y*sqrt(ln(1/y))
which solution is : y = exp(-(x+c))


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