dwsmith said:Can this be written as a system since it only has theta?
$$
U' = -\frac{mgb}{\sin^2\theta} - \frac{Mgb\cos\theta}{\sin^2\theta} = \frac{gb}{\sin^2\theta}(m - M\cos\theta).
$$
An ode, or ordinary differential equation, is a mathematical equation that describes how a variable changes over time. It involves a function and its derivatives, and can be used to model a wide range of phenomena in science and engineering.
To write an ode, you must first identify the variables and their derivatives that are involved in the system. Then, you can use mathematical operations and rules to express the relationships between these variables and their derivatives. Finally, you can solve the ode using various methods such as separation of variables, substitution, or numerical techniques.
Solving for theta in an ode means finding the value or values of theta that satisfy the differential equation. This involves manipulating the ode to isolate theta on one side of the equation and then using mathematical techniques to determine its value.
Solving for theta in ode systems allows us to understand the behavior and dynamics of the system. By finding the values of theta, we can determine the stability, equilibrium points, and other important properties of the system. This information is crucial in many scientific fields, including physics, chemistry, and biology.
Some common techniques for solving ode systems include separation of variables, substitution, and numerical techniques such as Euler's method or Runge-Kutta methods. Each method has its advantages and limitations, and the choice of technique depends on the specific system and its characteristics.