Coupled first order differential equations

In summary, the conversation discusses solving a problem involving the time-dependent Schrodinger equation for magnetic moment in time-dependent magnetic fields. The resulting equations are two coupled first order ODEs with imaginary parts, which can be solved using a nonlinear solver or by solving one ODE and substituting into the other.
  • #1
omni-impotent
12
0
I am trying to solve a problem (not homework, too old for that! lol!) which involves the time dependent schrodinger equation for magnetic moment in time-dependent magnetic fields. I end up with the following that needs to be solved:

x' = -i*(b*t-a*t^2)*x - i*c*y
y' = -i*c*x - i*(a*t^2-b*t)*y;

where i^2 = -1.

These look like 2 coupled 1st order ODE, but are in fact 4 coupled 1st order due to the imaginary parts. Any hints?
 
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  • #2
The easiest way is a nonlinear solver, like Mathematica, Maple or MATLAB. I guess you could even use Excel with the Solver nonlinear package.

I'm not a mathematician, but why can't you solve one ODE and substitute into the other one?
 

1. What are coupled first order differential equations?

Coupled first order differential equations are a set of two or more equations that describe the rate of change of multiple variables with respect to an independent variable. These equations are "coupled" because they are interdependent, meaning that the value of one variable affects the value of the others.

2. What are some real-world applications of coupled first order differential equations?

Coupled first order differential equations are used in many fields of science and engineering, such as physics, chemistry, biology, and economics. Some examples of their applications include modeling population growth, analyzing chemical reactions, and predicting the motion of celestial bodies.

3. How are coupled first order differential equations solved?

There are various methods for solving coupled first order differential equations, including separation of variables, substitution, and the use of matrices. Some equations may also be solved using numerical methods, such as Euler's method or the Runge-Kutta method.

4. What is the significance of initial conditions in solving coupled first order differential equations?

Initial conditions are values given for each variable at the starting point of the independent variable. They are crucial in solving coupled first order differential equations because they help determine the unique solution to the equations. Without initial conditions, the equations may have multiple solutions or no solution at all.

5. Can coupled first order differential equations have a closed-form solution?

In some cases, coupled first order differential equations may have a closed-form solution, meaning an equation that expresses the solution in terms of known functions. However, this is not always possible, and numerical methods may need to be used to approximate the solution.

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