Find an Autonomous ODE with Specified Properties

In summary: This solution has a divergence, not a maximum, though! I'm not sure that's what williamrand1 is looking for.
  • #1
williamrand1
21
0
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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  • #2
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.
 
  • #3
What do you think of y' = - y^(3/2) ?
 
  • #4
pasmith said:
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?
 
  • #5
JJacquelin said:
What do you think of y' = - y^(3/2) ?

Thanks JJ

Is there an exact solution to this?
 
  • #6
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
JJacquelin said:
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?
 
  • #8
Hi williamrand1 !

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

What is an autonomous ODE?

An autonomous ODE (ordinary differential equation) is a type of differential equation in which the independent variable does not explicitly appear. In other words, the rate of change of the dependent variable is only dependent on itself, rather than any external factors.

Why is it important to find an autonomous ODE with specified properties?

Finding an autonomous ODE with specified properties is important because it allows us to accurately model and understand systems in which the rate of change of a variable is only dependent on itself. This is particularly useful in fields such as physics, biology, and engineering.

What are some common properties that are specified when looking for an autonomous ODE?

Some common properties that are specified when looking for an autonomous ODE include the order of the equation, the type of equation (such as linear or nonlinear), the initial conditions, and any constraints on the solution.

How do you find an autonomous ODE with specified properties?

The process of finding an autonomous ODE with specified properties involves manipulating the general form of an autonomous ODE to fit the desired properties. This may involve using techniques such as substitution, integration, and separation of variables.

What are some applications of autonomous ODEs with specified properties?

Autonomous ODEs with specified properties have a wide range of applications, from modeling physical systems such as pendulums and circuits, to predicting population growth in biology, to optimizing processes in engineering. They are also essential in the development of mathematical models for complex systems.

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