Simultaneous differential equation of second order

In summary, the conversation is about a simultaneous differential equation of second order that describes the movement of a point mass around another point mass. The speaker is seeking a simpler and faster method to solve the equation, and suggests using polar coordinates instead of Cartesian coordinates. They later mention that they have successfully solved the equation using polar coordinates.
  • #1
player1_1_1
114
0
Hello
sorry for my English, i know its bad;)
I have a simultaneous differential equation of second order (moving of point mass around point mass M in beginning of cartesian system)
[tex]\begin{cases}\frac{\mbox{d}^2x}{\mbox{d}t^2}=-GMx\left(x^2+y^2\right)^{-\frac{3}{2}}\\ \frac{\mbox{d}^2y}{\mbox{d}t^2}=-GMy\left(x^2+y^2\right)^{-\frac{3}{2}}\end{cases}[/tex]
and I need to find how x and y changes depending on t.
solving by substitution takes very long time, so other method which allows to solve it in shorter and more simply way would be appreciated;)
i am not sure if this thread is good for this topic, please move it if its bad location
thanks for your help;)
 
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  • #2
Newton solved those equations in polar coordinates. I have never seen them solved in Cartesian coordinates.
 
  • #3
yeah, i came to it that it must be in polar system, i solved it and everything is good:)
 

FAQ: Simultaneous differential equation of second order

What is a simultaneous differential equation of second order?

A simultaneous differential equation of second order is a mathematical equation that involves two or more functions and their derivatives with respect to a common independent variable. This type of equation is commonly used in physics and engineering to model complex systems.

What is the difference between a simultaneous differential equation of second order and a single differential equation of second order?

The main difference is that a simultaneous differential equation involves multiple functions and their derivatives, while a single differential equation involves only one function and its derivatives. Additionally, a simultaneous differential equation can be solved for all of the functions simultaneously, while a single differential equation must be solved for one function at a time.

Can simultaneous differential equations of second order be solved analytically?

In some cases, yes, simultaneous differential equations can be solved analytically using methods such as separation of variables or substitution. However, for more complex systems, numerical methods may be necessary to find a solution.

What are some real-world applications of simultaneous differential equations of second order?

Simultaneous differential equations are commonly used in physics and engineering to model systems such as circuits, pendulums, and chemical reactions. They can also be used in economics to model supply and demand dynamics.

What techniques can be used to solve simultaneous differential equations of second order?

Some common techniques include separation of variables, substitution, power series solutions, and numerical methods such as Euler's method or Runge-Kutta methods. The specific technique used will depend on the complexity of the system and the desired level of accuracy.

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