Autonomous Second Order ODE

In summary, during a conversation about solving a general equation, an attached document was mentioned and a question was posed about understanding the provided solution. The solution involves defining a new variable and integrating to find the solution.
  • #1
muzialis
166
1
Hi All,

I was looking for the general solution of an equation as y(x)'' = f (y), and found the attached document on the web.

It presents the solution in a way which I am not sure I understand. I tried to look at the trivila example y'' = - y, solution y = sin (x), but I am struggling in obtaining this result with the provided solution.

If anybody could help, I would be the most obliged.

All the Best

Muzialis
 

Attachments

  • ode0301.pdf
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  • #2
Hi,

The trick is to define a new variable, y'=dy/dx=p(y(x)).
Now y''=dp/dy*y'=dp/dy*p=f(y).
Integrating now yields p^2=c+int(f(y))dy. The next step is to integrate dx=dy/p(y).

Mathador
 

1. What is an autonomous second order ODE?

An autonomous second order ODE, short for ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. It is called "autonomous" because the equation does not depend on the independent variable, and "second order" because it involves the second derivative of the function.

2. What is the difference between an autonomous and non-autonomous second order ODE?

As mentioned, an autonomous second order ODE does not depend on the independent variable, while a non-autonomous one does. This means that the coefficients in an autonomous ODE are constant, while in a non-autonomous ODE, they can vary with the independent variable.

3. What are some real-world applications of autonomous second order ODEs?

Autonomous second order ODEs have various applications in physics, engineering, and other scientific fields. They can be used to model the motion of a pendulum, the behavior of a spring-mass system, or the oscillations of an electrical circuit, to name a few examples.

4. How do you solve an autonomous second order ODE?

Solving an autonomous second order ODE involves finding a function that satisfies the equation, given some initial conditions. This can be done analytically, using methods such as separation of variables or the characteristic equation, or numerically, through approximation methods like Euler's method or the Runge-Kutta method.

5. What are the challenges in solving autonomous second order ODEs?

The main challenge in solving autonomous second order ODEs is finding an exact analytical solution. In most cases, this is not possible, and numerical methods must be used. Additionally, some ODEs may have multiple solutions or no solutions at all, and determining the appropriate solution can be challenging.

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