## Autonomous ODE

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

## Autonomous ODE

 Quote by 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.
Thanks pasmith

Could you explain why it is not possible?

 Quote by JJacquelin What do you think of y' = - y^(3/2) ?
Thanks JJ

Is there an exact solution to this?

 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

Recognitions:
Homework Help
 Quote by 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
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?

 Hi williamrand1 ! Then, what about this one : y' = -2y*sqrt(ln(1/y)) which solution is : y = exp(-(x+c)²)