Symbolic ODE Solving

While not all ODEs have an analytical solution, symbolic ODE solving is provided by Symbolics.jl for subsets of cases for which known analytical solutions can be obtained. These expressions can then be merged with other techniques in order to accelerate code or gain a deeper understanding of real-world systems.

Merging Symbolic ODEs with Numerical Methods: ModelingToolkit.jl

If you are looking to merge Symbolics.jl manipulations with numerical solvers such as DifferentialEquations.jl, then we highly recommend checking out the ModelingToolkit.jl system. It represents systems of differentible-algebraic equations (DAEs, and extension to ODEs) for which a sophisticated symbolic analysis pipeline is used to generate highly efficient code. This mtkcompile pipeline makes use of all tricks from Symbolics.jl and many that are more specific to numerical code generation, such as Pantelides index reduction and tearing of nonlinear systems, and it will analytically solve subsets of the ODE system as finds possible. Thus if you are attempting to use Symbolics to pre-solve some parts of an ODE analytically, we recommend allowing ModelingToolkit.jl to do this optimization.

Note that if ModelingToolkit is able to analytically solve the equation, it will give an ODEProblem where prob.u0 === nothing, and then running solve on the ODEProblem will give a numerical ODESolution object that on-demand uses the analytical solution to generate any plots or other artifacts. The analytical solution can be investigated symbolically using observed(sys).

Symbolically Solving ODEs

Currently there is no native symbolic ODE solver. Though there are bindings to SymPy

Note

This area is currently under heavy development. More solvers will be available in the near future.

SymPy

Symbolics.sympy_ode_solve — Function

sympy_ode_solve(expr, func, var)

Solves ODE expr = 0 for function func w.r.t. var using SymPy.

Arguments

expr: Symbolics expression representing ODE (set to 0).
func: Symbolics function (e.g., f(x)).
var: Independent Symbolics variable.

Returns

Symbolics solution(s).

Example

@variables x
@syms f(x)
expr = Symbolics.Derivative(f, x) - 2*f
sol = sympy_ode_solve(expr, f, x)  # Returns C1*exp(2*x)

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