Since the independent variable, which I will call "t", does not appear explicitely in the equation, this is a candidate for "quadrature". Let [itex]y= dx/dt[/itex]. Then [itex]d^2x/dt^2= (dy/dx)(dx/dt)[/itex] by the chain rule. But since [itex]y= dx/dt[/itex], that is [itex]d^2x/dt^2= y dy/dx[/itex] and the equation converts to the first order equation, for y as a function of x, [itex]y dy/dx= -cos(x)[/itex] which is [itex]y dy= -cos(x)dx[/itex]. Integrating, [itex](1/2)y^2= sin(x)+ C[/itex]. Now we have [itex]y^2= -2 sin(x)+ 2C[/itex] or [itex]y= dx/dt= \sqrt{2C- 2sin(x)}[/itex] so [itex]dx= \sqrt{2C- 2sin(x)}dx[/itex].Reid said:Homework Statement
I can not determine the solution to the diff. eq.
Homework Equations
[tex]\ddot{x}+k \cos{x}=0[/tex].
The constant k is postive.
The Attempt at a Solution
I tried solving it with the methods I know and it all ended up in a big mess. I am just not used to the second term, \cos. Doe's anyone know what to do?
An ODE (Ordinary Differential Equation) is a mathematical equation that relates a function to its derivatives. It is used to model many physical phenomena, such as motion, heat transfer, and chemical reactions.
An ODE is considered "weird" if it contains non-traditional functions, such as cosine, in its equation. These functions can make the solution more complicated and difficult to obtain.
Cosine is often used in ODEs to model periodic behavior, such as in oscillations or vibrations. It can also be used to represent the amplitude or phase of a system.
In general, ODEs with cosine cannot be solved analytically (i.e. using algebraic operations). However, some special cases may have exact solutions, and numerical methods can also be used to approximate solutions.
ODEs with cosine can be used to model a wide range of physical phenomena, such as pendulums, electrical circuits, and population dynamics. They are also important in fields such as engineering, physics, and biology for understanding and predicting real-world behavior.