Homogeneous 2nd Order DE from spring pendulum

In summary, The speaker is taking a Classical Mechanics course and studying the Lagrangian equation. They attempted to find the motion equations for a pendulum with a mass hanging from a spring and encountered differential equations that they don't know how to solve. They are seeking information on the name or method for solving these equations.
  • #1
DavitosanX
9
1
I'm currently taking a Classical Mechanics course, and we're studying the lagrangian equation. After a few exercises, I thought I'd try to come up with the motion equations for a pendulum where the mass hangs from a spring. The resulting differential equations take a form that I don't really know how to solve. They take the form:

y'' + x*y + g*cos(y) + k/m = 0

and

y'' - (g*sin(y))/x = 0

I would just like to know if these have a particular name, or maybe a specific method to solve them. Thanks for the help!
 
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  • #2
You should check your derivation. The equations that you have don't look right.
 

Related to Homogeneous 2nd Order DE from spring pendulum

1. What is a homogeneous 2nd order DE?

A homogeneous 2nd order differential equation (DE) is an equation that involves a second derivative of an unknown function, and all terms in the equation are homogeneous, meaning they are of the same degree. In the context of a spring pendulum, this type of DE can be used to model the oscillations of the pendulum.

2. How is a homogeneous 2nd order DE used to model a spring pendulum?

A spring pendulum can be modeled using a homogeneous 2nd order DE by considering the forces acting on the pendulum, such as gravity and the restoring force of the spring. By using the equation F=ma and applying Newton's Second Law, we can derive a homogeneous 2nd order DE that represents the motion of the pendulum.

3. What is the solution to a homogeneous 2nd order DE for a spring pendulum?

The solution to a homogeneous 2nd order DE for a spring pendulum is a sinusoidal function, since the motion of the pendulum is an oscillation. The specific form of the solution will depend on the initial conditions of the pendulum, such as its initial displacement and velocity.

4. How does the mass and spring constant affect the homogeneous 2nd order DE for a spring pendulum?

The mass and spring constant will affect the coefficients in the homogeneous 2nd order DE for a spring pendulum. A larger mass will result in a larger coefficient for the acceleration term, while a larger spring constant will result in a larger coefficient for the displacement term.

5. Can a homogeneous 2nd order DE be solved analytically for a spring pendulum?

Yes, a homogeneous 2nd order DE for a spring pendulum can be solved analytically using techniques such as separation of variables or the method of undetermined coefficients. However, in some cases, numerical methods may be needed to find an approximate solution.

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