- #1

Logarythmic

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How do I solve

[tex]\ddot{\theta} = c \cos{\theta}[/tex]?

[tex]\ddot{\theta} = c \cos{\theta}[/tex]?

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- Thread starter Logarythmic
- Start date

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

- #1

Logarythmic

- 281

- 0

How do I solve

[tex]\ddot{\theta} = c \cos{\theta}[/tex]?

[tex]\ddot{\theta} = c \cos{\theta}[/tex]?

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- #2

Integral

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Science Advisor

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[tex] \ddot{\theta} = c [/tex]

- #3

- 13,356

- 3,379

Logarythmic said:How do I solve

[tex]\ddot{\theta} = c \cos{\theta}[/tex]?

Multiply by [itex] 2\dot{\theta} [/itex] and then notice that you get

[tex] \frac{d}{dt}\dot{\theta}^{2} = c\frac{d}{dt}{\sin\theta} [/tex]

The rest is easy.

Daniel.

- #4

Logarythmic

- 281

- 0

Is it? =P

[tex]\dot{\theta}^2 = c_1 \sin{\theta} + c_2[/tex]?

[tex]\dot{\theta}^2 = c_1 \sin{\theta} + c_2[/tex]?

- #5

- 13,356

- 3,379

Yes, now separate variables and integrate.

Daniel.

Daniel.

- #6

HallsofIvy

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Homework Helper

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"The rest is easy"!

- #7

Logarythmic

- 281

- 0

[tex]\int \frac{d\theta}{\sqrt{c - \frac{3g}{L}sin{\theta}}}[/tex]

to solve. Any tip?

- #8

- 13,356

- 3,379

Yes, use the web integrator from Mathematica's website:http://integrals.wolfram.com/index.jsp

The result is

The result is

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.

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.

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.

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.

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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