A Property of an Autonomous ODE

In summary, the question is how to prove that if y(t)=sin(t) is a solution of an autonomous ODE f(y,y',...,y^(n))=0, then x(t)=cos(t) is also a solution. The person asking the question is unsure how to proceed since the equation includes all derivatives of y, while in the case of a pendulum, only y and y'' are present with a special functional dependence between them. They are also considering using the relationship cos(x) = sin(x + pi / 2) to help solve the problem.
  • #1
littleHilbert
56
0
Hi!

I wonder how to prove that if y(t)=sin(t) solves an autonomous ODE f(y,y',...,y^(n))=0, then x(t)=cos(t) is also a solution.

I mean I'm a bit distracted by the fact that all derivatives of y are present here. For example in the equation for a pendulum there are just y and y'' and a special functional dependence between them and that's why it works, isn't it?

In the above case I know almost nothing about the function f. So how to proceed?
 
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  • #2
Maybe you can use that cos(x) = sin(x + pi / 2)?
 

FAQ: A Property of an Autonomous ODE

1. What is an Autonomous ODE?

An Autonomous ODE, or Ordinary Differential Equation, is a type of mathematical equation that describes the relationship between a function and its derivatives. It is "autonomous" because it does not depend on an independent variable, meaning the function and its derivatives are solely dependent on each other.

2. How are Autonomous ODEs used in science?

Autonomous ODEs are used in many different scientific fields, including physics, chemistry, biology, and engineering. They are used to model and analyze systems that change over time, such as population growth, chemical reactions, and mechanical systems.

3. What is the difference between an Autonomous ODE and a Non-Autonomous ODE?

The main difference between an Autonomous ODE and a Non-Autonomous ODE is that an Autonomous ODE does not depend on an independent variable, while a Non-Autonomous ODE does. This means that the solutions to Autonomous ODEs are constant over time, while solutions to Non-Autonomous ODEs can vary.

4. How do you solve an Autonomous ODE?

The most common method for solving Autonomous ODEs is by separation of variables. This involves isolating the dependent variable on one side of the equation and the independent variable and its derivatives on the other side, then integrating both sides. Other methods include substitution, series solutions, and numerical methods.

5. What are some real-world applications of Autonomous ODEs?

Autonomous ODEs have many real-world applications, such as predicting population growth, modeling chemical reactions, analyzing electrical circuits, and understanding the motion of objects. They are also used in the fields of economics, ecology, and climate science to model and predict complex systems.

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