Find an Autonomous ODE with Specified Properties

[williamrand1]

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...

 

[pasmith]

This is impossible.

Suppose the maximum is at t = t_0. Then there exist t_1 < t_0 < t_2 such that x(t_1) = x(t_2), but \dot x(t_1) = -\dot x(t_2). 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]

What do you think of y' = - y^(3/2) ?

 

[williamrand1]

This is impossible.

Suppose the maximum is at t = t_0. Then there exist t_1 < t_0 < t_2 such that x(t_1) = x(t_2), but \dot x(t_1) = -\dot x(t_2). 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]

What do you think of y' = - y^(3/2) ?

Thanks JJ

Is there an exact solution to this?

 

[JJacquelin]

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)²

 

[Mute]

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)²

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]

Hi williamrand1 !

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