Autonomous ODE

1. Jan 23, 2013

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...

2. Jan 23, 2013

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.

3. Jan 24, 2013

JJacquelin

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

4. Jan 24, 2013

williamrand1

Thanks pasmith

Could you explain why it is not possible?

5. Jan 24, 2013

williamrand1

Thanks JJ

Is there an exact solution to this?

6. Jan 24, 2013

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

7. Jan 24, 2013

Mute

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?

8. Jan 25, 2013

JJacquelin

Hi williamrand1 !

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

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