Unique solution of 1st order autonomous, homogeneous DE

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In summary, the topic of discussion is the general form of 1st order autonomous, homogeneous differential equations. The unique solution for such equations is x(t)=e^{at}x(t_{0}). The conversation then delves into clarifying the meaning of x(t) and how to calculate its derivative and integral. The expert summarizer advises to integrate both sides of the equation and notes that x is the only variable on the left side.
  • #1
Pietair
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Hello,

1st order autonomous, homogeneous differential equation have the general form:
[tex]\dot{x}(t)=ax(t)[/tex]

It can be shown that the unique solution is always:

[tex]x(t)=e^{at}x(t_{0})[/tex]

I don't get this, could anyone help me?

Thanks!
 
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  • #2
Have you tried at all? What is the derivative of [itex]e^{at}[/itex]. Since [itex]x(t_0)[/itex] is a constant, what is the derivative of [itex]e^{at}x(t_0)[/itex]?
 
  • #3
Thank you for your answer.

I can work it out when x(t) = x, but this is not the case, is it?
 
  • #4
x(t) is supposed to be a function of t. What do you mean by x(t)=x?
 
  • #5
How can I calculate the integral of x(t) when I don't know the corresponding function? x(t) can equal (t^2) or (t-3) and so on, right?
 
  • #6
Will you please answer my questions? Do you know what the derivative of [itex]e^{at}[/itex] is? Do you know what the derivative of [itex]e^{at}x(t_0)[/itex] is?
 
  • #7
Pietair said:
How can I calculate the integral of x(t) when I don't know the corresponding function? x(t) can equal (t^2) or (t-3) and so on, right?
You can't and you don't want to.

If [tex]\frac{dx}{dt}= ax[/tex] then
[tex]\frac{dx}{x}= adt[/tex]

Integrate both sides of
[tex]\int \frac{dx}{x}= \int a dt[/tex]

Note that on the left you have NO "t". The only variable is x.
 

1. What is a first-order autonomous, homogeneous differential equation?

A first-order autonomous, homogeneous differential equation is a mathematical equation that involves a single independent variable, its derivative, and no other variables. It is also known as a first-order ordinary differential equation (ODE) and is considered autonomous because the independent variable does not appear explicitly in the equation. Additionally, the equation is homogeneous because all terms have the same degree.

2. What does it mean for a first-order autonomous, homogeneous differential equation to have a unique solution?

Having a unique solution means that there is only one solution that satisfies the given initial conditions. This means that for any given value of the independent variable, there is only one corresponding value for the dependent variable that satisfies the equation. In other words, there are no other possible solutions for the given initial conditions.

3. How do you solve a first-order autonomous, homogeneous differential equation?

To solve a first-order autonomous, homogeneous differential equation, one can use the method of separation of variables. This involves separating the variables and integrating both sides of the equation. After solving for the constant of integration, the resulting equation will be the general solution. The specific solution can then be found by substituting the given initial conditions into the general solution.

4. Can a first-order autonomous, homogeneous differential equation have more than one solution?

No, a first-order autonomous, homogeneous differential equation can only have a unique solution. This is because the equation is autonomous and homogeneous, meaning that there is only one possible solution for a given set of initial conditions.

5. What are the applications of first-order autonomous, homogeneous differential equations?

First-order autonomous, homogeneous differential equations have many applications in various fields, including physics, chemistry, biology, economics, and engineering. They are used to model and analyze various phenomena, such as population growth, chemical reactions, and electrical circuits. Additionally, they are used in control systems and machine learning algorithms.

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