Solving Simultaneous Equations

In summary, to obtain expressions for X(t) and Y(t), the equations [1] and [2] can be rearranged and solved. Alternatively, the equations can be differentiated and used to express dY/dt in terms of X, resulting in a second order differential equation. This is similar to the equations for simple harmonic motion.
  • #1
masej
1
0

Homework Statement



I need to solve these two equations to obtain expressions for [itex]X(t)[/itex] and [itex]Y(t)[/itex]

Homework Equations



[1]. [tex]\frac{d(X(t))}{dt} = Q \cdot Y(t)[/tex]

[2]. [tex]\frac{d(Y(t))}{dt} = -Q \cdot X(t)[/tex]

The Attempt at a Solution



Perhaps rearrange equation [1] to get in terms of [itex]Y(t)[/itex] then input this into expression into equation [2] to get equation ([3]) just with [itex]X(t)[/itex] terms. Then solve [3] to find [itex]X(t)[/itex] and input this back into equation [1].

.. hows that? It gets rather messy :frown: .
 
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  • #2
Fairly obviously you can just divide and get dY/dX independent of t, get a solution, use that to express dX/dt in terms of X alone.

Or you could differentiate (1) and RHS is dY/dt for which by eq. 2 you can express in terms of X and so get a 2nd order d.e.

Or you can see almost any textbook of math (or phys) that covers de's and get it more generally, if more longwindedly (I think it is easier to solve yrself and easier to read the books if you have done something your self).

(In fact this is nothing but the equations for simple harmonic motion - special case or special units isn't it, X displacement, Y momentum?)
 
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1. What are simultaneous equations?

Simultaneous equations are a set of two or more equations with multiple variables that have a common solution. This means that the values of the variables must satisfy all of the equations at the same time.

2. What is the process for solving simultaneous equations?

The process for solving simultaneous equations involves using algebraic methods such as substitution or elimination to find the values of the variables that satisfy all of the equations in the system.

3. Can simultaneous equations have more than two variables?

Yes, simultaneous equations can have any number of variables, but the number of equations in the system must be equal to the number of variables in order to have a unique solution.

4. How do you know if a set of simultaneous equations has a solution?

A set of simultaneous equations has a solution if the equations are consistent, meaning that there is at least one set of values for the variables that satisfies all of the equations in the system. If the equations are inconsistent, there is no solution, and if they are dependent, there are infinite solutions.

5. Are there any shortcuts or tricks for solving simultaneous equations?

Yes, there are various shortcuts and tricks for solving simultaneous equations, such as the elimination method, substitution method, and graphing method. However, it is important to understand the underlying algebraic concepts and methods in order to effectively solve any type of simultaneous equations.

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