solvers.rst

Solvers

This section covers equations solving.

Note

Any expression in input, that not in an ~diofant.core.relational.Eq is automatically assumed to be equal to 0 by the solving functions.

Algebraic Equations

The main function for solving algebraic equations is ~diofant.solvers.solvers.solve.

When solving a single equation, the output is a list of the solutions.

>>> solve(x**2 - x) [{x: 0}, {x: 1}]

If no solutions are found, an empty list is returned.

>>> solve(exp(x)) []

~diofant.solvers.solvers.solve can also solve systems of equations.

>>> solve([x - y + 2, x + y - 3]) [{x: 1/2, y: 5/2}] >>> solve([x*y - 7, x + y - 6]) ⎡⎧ ___ ___ ⎫ ⎧ ___ ___ ⎫⎤ ⎢⎨x: - ╲╱ 2 + 3, y: ╲╱ 2 + 3⎬, ⎨x: ╲╱ 2 + 3, y: - ╲╱ 2 + 3⎬⎥ ⎣⎩ ⎭ ⎩ ⎭⎦

Each solution reported only once:

>>> solve(x*3 - 6x*2 + 9x) [{x: 0}, {x: 3}]

To get the solutions of a polynomial including multiplicity use ~diofant.polys.polyroots.roots.

>>> roots(x*3 - 6x*2 + 9x) {0: 1, 3: 2}

Recurrence Equations

To solve recurrence equations, use ~diofant.solvers.recurr.rsolve. First, create an undefined function by passing cls=Function to the ~diofant.core.symbol.symbols function.

>>> f = symbols('f', cls=Function)

We can call f(x), and it will represent an unknown function application.

Note

From here on in this tutorial we assume that these statements were executed:

>>> from diofant import * >>> a, b, c, d, t, x, y, z = symbols('a:d t x:z') >>> k, m, n = symbols('k m n', integer=True) >>> f, g, h = symbols('f:h', cls=Function) >>> init_printing(pretty_print=True, use_unicode=True)

As for algebraic equations, the output is a list of dict's

>>> rsolve(f(n + 1) - 3*f(n) - 1) ⎡⎧ n 1⎫⎤ ⎢⎨f: n ↦ 3 ⋅C₀ - ─⎬⎥ ⎣⎩ 2⎭⎦

The arbitrary constants in the solutions are symbols of the form C0, C1, and so on.

Differential Equations

To solve the differential equation

>>> Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))

2

d d

f(x) - 2⋅──(f(x)) + ───(f(x)) = sin(x)

dx 2

dx

Note

Derivatives of the unknown function f(x) are unevaluated.

we would use

>>> dsolve(_)

x cos(x)

f(x) = ℯ ⋅(C₁ + C₂⋅x) + ──────

2

~diofant.solvers.ode.dsolve can also solve systems of equations, like ~diofant.solvers.solvers.solve.

>>> dsolve([f(x).diff(x) - g(x), g(x).diff(x) - f(x)]) ⎡ x -x x -x ⎤ ⎣f(x) = ℯ ⋅C₂ - ℯ ⋅C₁, g(x) = ℯ ⋅C₂ + ℯ ⋅C₁⎦