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Autonomous ODEby williamrand1
Tags: autonomous 
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#1
Jan2313, 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... 


#2
Jan2313, 04:30 PM

HW Helper
Thanks
P: 1,004

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 secondorder autonomous ODE, as should be obvious from the fact that you want to satisfy two boundary conditions. 


#3
Jan2413, 01:06 AM

P: 759

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



#4
Jan2413, 11:16 AM

P: 21

Autonomous ODE
Could you explain why it is not possible? 


#5
Jan2413, 11:18 AM

P: 21

Is there an exact solution to this? 


#6
Jan2413, 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/(xC) y = 4/(xC)² 


#7
Jan2413, 10:40 PM

HW Helper
P: 1,391

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
Jan2513, 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)²) 


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