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Symbolic ODE solver #1572

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LordOfFrogs
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Created basic solvers for 1st order separable equations and linear continuous dynamical systems with real eigenvalues. I'm sure there are better ways to do a lot of this, as I am still familiarizing myself with Symbolics.jl and Julia in general.

Implemented symbolic eigenvalue and eigenvector decomposition because LinearAlgebra.jl functions produced floating-point inaccuracies that messed up rationals.

Also, not currently incorporated into Symbolics.jl yet, just a standalone file added.

LordOfFrogs added 19 commits June 25, 2025 22:16
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Created basic solvers for 1st order separable eqs and linear systems
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added rationalization to inverted matrix in solve_linear_system
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integrated into Symbolics, restructured files
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added unit tests
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added display functionality for LinearODE
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implemented solver for homogeneous solutions (of degree 5 or less wit…
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…h no repeated roots)
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added solving via integrating factor (1st order ODEs)
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added handling of repeated characteristic roots
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implemented exponential response formula / resonant response formula
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added solving particular solutions when q(t) includes sin or cos and …
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…an exponential
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added handling of complex characteristic roots
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slight refactoring and improved documentation
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added method of undetermined coefficients
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added expression parsing
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updated cases handled
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added IVP
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added method to solve Clairaut's equation
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implemented solving bernoulli equations
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updated documentation and fixed test imports
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@LordOfFrogs LordOfFrogs changed the title [WIP] Symbolic ODE solver Symbolic ODE solver Jul 18, 2025
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After updating the documentation, this should be ready to merge. It now includes methods for solving symbolic ODEs via:

  • first-order integrating factor
  • constant coefficient homogeneous solutions (can handle repeated and complex characteristic roots)
  • exponential and resonant response formula particular solutions (for any linear combination of exp, sin, cos, or exp times sin or cos (e.g. e^2t * cos(-t) + e^-3t + sin(5t)))
  • method of undetermined coefficients particular solutions
  • linear combinations of above particular solutions
  • Clairaut's equation
  • Bernoulli equations

Additionally, it can now parse symbolic expressions into LinearODE or use them directly to solve.

Solving continuous dynamical systems with systems.jl still appears to be finicky at times, due to floating-point inaccuracies and the handling of complex numbers.

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