# Coupled 2nd-Order Non-linear ODEs

• cepheid
In summary, the conversation discusses solving two equations involving differentiation with respect to a parameter. The first equation is tackled using the chain rule, and the resulting equation is integrated to obtain a proportionality relationship for \dot{\phi}. This result is then substituted into the second equation, but the final equation obtained is not clear. Further insight is needed to solve the equations completely.
cepheid
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## Homework Statement

I'm trying to solve the equations:

$$\ddot{\phi} + 2\left(\frac{\cos \theta}{\sin \theta}\right) \dot{\theta}\dot{\phi} =0$$​

and

$$\ddot{\theta} - \sin \theta \cos \theta \dot{\phi^2} =0$$​

for $\theta(\lambda), \phi(\lambda)$ where the dots represent differentiation w.r.t the parameter $\lambda$.

## Homework Equations

Chain rule, other differentiation tricks

## The Attempt at a Solution

I've tried to tackle the first equation by writing:

$$\frac{\cos \theta}{\sin \theta} \dot{\theta} = \frac{1}{\sin \theta} \frac{d}{d\theta}(\sin \theta) \dot{\theta} = \frac{d}{d\lambda}[\ln (\sin \theta) ]$$​

I'm wondering if it's correct to do that. It seems to follow from the chain rule, but I'm not sure. If so, then the equation becomes:

$$\frac{\ddot{\phi}}{\dot{\phi}} = -2\frac{d}{d\lambda}[\ln (\sin \theta) ]$$

$$\frac{1}{\dot{\phi}}\frac{d \dot{\phi}}{d\lambda} = -2\frac{d}{d\lambda}[\ln (\sin \theta) ]$$

$$\frac{d \ln ( \dot{\phi})}{d\lambda} = -2\frac{d}{d\lambda}[\ln (\sin \theta) ]$$​

Then I integrated both sides w.r.t. lambda:

$$\ln ( \dot{\phi}) = - 2\ln (\sin \theta) + \mathrm{~const.}$$

$$\dot{\phi} \propto \frac{1}{\sin^2 \theta}$$​

Then I tried substituting this result into the second ODE, but it gave me:

$$\ddot{\theta} \propto \frac{\cos \theta}{\sin^3 \theta}$$​

which I don't know what to do with.

Last edited:
Just bumping the thread. Any insight you might have, esp. regarding the last line, would be most helpful.

## 1. What is a coupled 2nd-order non-linear ODE?

A coupled 2nd-order non-linear ODE (ordinary differential equation) is a mathematical equation that describes the relationship between two variables, where the second derivative of both variables is present and at least one of the variables is non-linear. This means that the equation is not a straight line and cannot be solved using traditional methods.

## 2. How is a coupled 2nd-order non-linear ODE solved?

There is no general method for solving coupled 2nd-order non-linear ODEs. However, there are several techniques that can be used depending on the specific equation, such as substitution, linearization, or numerical methods like Euler's method or Runge-Kutta methods.

## 3. What are some real-life applications of coupled 2nd-order non-linear ODEs?

Coupled 2nd-order non-linear ODEs are commonly used in physics, engineering, and biology to model complex systems. Some examples include the motion of a double pendulum, the behavior of electrical circuits, and the dynamics of chemical reactions.

## 4. How do you know if a system can be described by a coupled 2nd-order non-linear ODE?

If the relationship between two variables in a system involves the second derivatives of both variables and at least one variable is non-linear, then it can be described by a coupled 2nd-order non-linear ODE. Additionally, if the system is complex and the relationship between the variables cannot be described by a simple linear equation, it is likely that a coupled 2nd-order non-linear ODE is needed.

## 5. What are some challenges of working with coupled 2nd-order non-linear ODEs?

Coupled 2nd-order non-linear ODEs can be very difficult to solve analytically, meaning that exact solutions are not always possible. This requires the use of numerical methods, which may introduce errors. Additionally, the behavior of non-linear systems can be unpredictable and chaotic, making it challenging to accurately model and predict outcomes.

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