Solve Differential Equation: \ddot{\theta} = c \cos{\theta}

• Logarythmic
In summary, to solve the equation \ddot{\theta} = c \cos{\theta} numerically, one can use Runga Kutta or for small angles, approximate cosine as 1 and solve for \ddot{\theta} = c. By multiplying by 2\dot{\theta} and noticing that it becomes \frac{d}{dt}\dot{\theta}^{2} = c\frac{d}{dt}{\sin\theta}, one can easily integrate and separate variables to obtain the integral \int \frac{d\theta}{\sqrt{c - \frac
Logarythmic
How do I solve

$$\ddot{\theta} = c \cos{\theta}$$?

Numerically. Try Runga Kutta, or for small angles Cos ( $\theta$) ~ 1 so solve

$$\ddot{\theta} = c$$

Logarythmic said:
How do I solve

$$\ddot{\theta} = c \cos{\theta}$$?

Multiply by $2\dot{\theta}$ and then notice that you get

$$\frac{d}{dt}\dot{\theta}^{2} = c\frac{d}{dt}{\sin\theta}$$

The rest is easy.

Daniel.

Is it? =P

$$\dot{\theta}^2 = c_1 \sin{\theta} + c_2$$?

Yes, now separate variables and integrate.

Daniel.

"The rest is easy"!

Ok, so now I got the integral

$$\int \frac{d\theta}{\sqrt{c - \frac{3g}{L}sin{\theta}}}$$

to solve. Any tip?

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1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to describe how a quantity changes over time or space, and is often used in various fields of science, including physics, engineering, and economics.

2. What does the symbol \ddot{\theta} represent in this equation?

The symbol \ddot{\theta} represents the second derivative of the function \theta with respect to time. This means that it represents the rate of change of the rate of change of the function \theta over time.

3. What does the constant "c" represent in this equation?

The constant "c" represents the coefficient of the cosine function. It determines the strength of the cosine term in the equation and can affect the behavior of the solution to the differential equation.

4. How do you solve this differential equation?

To solve this differential equation, you can use various techniques such as separation of variables, substitution, or integrating factors. The method used depends on the form of the equation and the techniques you are familiar with.

5. What is the significance of solving this differential equation?

Solving this differential equation can help us understand the behavior of a system or phenomenon that is described by the equation. It can also be used to predict future values of the function \theta and analyze the effects of different parameters on the solution.

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