Coupled first order differential equations

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SUMMARY

The discussion centers on solving a system of coupled first-order differential equations derived from the time-dependent Schrödinger equation for magnetic moments in time-dependent magnetic fields. The equations presented are x' = -i*(b*t-a*t^2)*x - i*c*y and y' = -i*c*x - i*(a*t^2-b*t)*y, which represent four coupled first-order ODEs due to their imaginary components. The recommended tools for solving these equations include Mathematica, Maple, MATLAB, and even Excel with the Solver nonlinear package. The user expresses confusion about the possibility of solving one ODE and substituting it into the other.

PREREQUISITES
  • Understanding of first-order ordinary differential equations (ODEs)
  • Familiarity with complex numbers and their properties
  • Knowledge of the time-dependent Schrödinger equation
  • Experience with numerical solvers like Mathematica, Maple, or MATLAB
NEXT STEPS
  • Explore the use of Mathematica for solving coupled differential equations
  • Learn about the numerical methods for solving ODEs in MATLAB
  • Investigate the capabilities of Maple for handling complex systems of equations
  • Study the application of Excel's Solver for nonlinear problem-solving
USEFUL FOR

Mathematicians, physicists, and engineers working with differential equations, particularly those interested in quantum mechanics and magnetic fields.

omni-impotent
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I am trying to solve a problem (not homework, too old for that! lol!) which involves the time dependent Schrödinger 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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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?
 

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