Second order DE with Sine function

In summary, the conversation discusses a problem with a second order differential equation involving a sine function and a constant. The person has tried to solve it using DSolve in Mathematica but has encountered an error. They ask for guidance and mention the possibility of using the small angle approximation. Another person recommends a book on non-linear equations which includes a step-by-step solution for the non-linear pendulum. The original person is still unable to find a solution and asks for hints or notes to help them solve the equation. The conversation also briefly discusses another equation that may be related.
  • #1
Figaro
103
7
I have this second order differential equation but I'm stumped as to how to solve this since the zeroth order term has a Sine function in it and the variable is embedded.

##\ddot y(t) + 3H (1+Q) \dot y(t) -m^2 f \sin(\frac{y(t)}{f}) = 0##

##H~##, ##~Q~##, ##~m~##, and ##~f~## are just constants.

I even tried to use DSolve in mathematica but there is an error. How do I solve this? Can anyone guide me with this problem?
 
Last edited:
Physics news on Phys.org
  • #2
Google damped pendulum differential equation
[edit] and there's always this
 
  • #3
BvU said:
Google damped pendulum differential equation
[edit] and there's always this
Are you pertaining to the small angle approximation? I know that is a possibility but the problem is that ##\theta = \frac{y[t]}{f} \approx 1##.
 
  • #4
Small angle is the first approach. The article in the link discusses the full equation in section 3.
 
  • #5
  • #6
aheight said:
If you're interested in non-linear equations, I recommend this book which includes a step-by step solution of the non-linear pendulum:

http://store.doverpublications.com/0486609715.html
I still can't find the solution to my given equation, can you kindly give me a hint/note onto where I should look?
 
  • #7
Figaro said:
I still can't find the solution to my given equation, can you kindly give me a hint/note onto where I should look?

I was referring to the equation ## y''=k\sin(y)##. That's not yours but if you first study how this one is solved exactly in terms of elliptic functions, then perhaps you can adapt the method to yours. It's been a while that I've studied it (using the book I quoted) and I no longer have the book and don't recall exactly how it's done.
 

Related to Second order DE with Sine function

1. What is a second order DE with sine function?

A second order differential equation (DE) with sine function is a type of mathematical equation that includes a second derivative of a variable and a sine function. It can be written in the form of y'' + a*y' + b*y = c*sin(x), where y is the dependent variable, a and b are constants, and c is the coefficient of the sine function.

2. What is the general solution to a second order DE with sine function?

The general solution to a second order DE with sine function is y(x) = c1*cos(x) + c2*sin(x) + y0, where c1 and c2 are constants determined by initial conditions, and y0 is the particular solution.

3. What is the difference between a homogeneous and non-homogeneous second order DE with sine function?

In a homogeneous second order DE with sine function, the right-hand side of the equation is equal to 0, while in a non-homogeneous equation, the right-hand side is a non-zero constant. This results in different solutions for the two types of equations.

4. How do you solve a second order DE with sine function using the method of undetermined coefficients?

The method of undetermined coefficients involves finding a particular solution to a non-homogeneous DE with sine function by assuming a solution of the form y = Asin(x) + Bcos(x), where A and B are constants. This particular solution is then added to the general solution of the corresponding homogeneous equation.

5. Can a second order DE with sine function be solved analytically?

Yes, a second order DE with sine function can be solved analytically using various methods such as variation of parameters or Laplace transforms. However, for more complex equations, numerical methods may be needed to obtain an approximate solution.

Similar threads

I Converting Second Order ODE to Hypergeometric Function
    • Differential Equations
Replies
7
Views
2K
I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
2K
I Second order non-homogeneous linear ordinary differential equation
    • Differential Equations
Replies
2
Views
2K
I Solve second order linear differential equation
    • Differential Equations
Replies
7
Views
1K
I How do I solve this first order second degree differential equation?
    • Differential Equations
Replies
7
Views
3K
A FEM basis polynomial order and the differential equation order
    • Differential Equations
Replies
1
Views
1K
I How to solve this second order differential equation
    • Differential Equations
Replies
10
Views
2K
I Solving Differential Equation Using Reduction of Order
    • Differential Equations
Replies
2
Views
1K
I Question about solving linear first order non-homogeneous ODEs
    • Differential Equations
Replies
3
Views
850
I On sub and super solutions: Teschl and others
    • Differential Equations
Replies
5
Views
709

Hot Threads

Recent Insights

Back
Top