Solve an Equation Algebraically¶
Use SymPy to solve an equation algebraically (symbolically). For example, solving \(x^2 = y\) for \(x\) yields \(x \in \{\sqrt{y},\sqrt{y}\}\).
Alternatives to consider¶
SymPy can also solve many other types of problems including sets of equations.
Some equations cannot be solved algebraically (either at all or by SymPy), so you may have to
solve your equation numerically
instead.
Solving Functions¶
There are two highlevel functions to solve equations, solve()
and
solveset()
. Here is an example of each:
>>> from sympy.abc import x, y
>>> from sympy import solve
>>> solve(x**2  y, x, dict=True)
[{x: sqrt(y)}, {x: sqrt(y)}]
>>> from sympy import solveset
>>> from sympy.abc import x, y
>>> solveset(x**2  y, x)
{sqrt(y), sqrt(y)}
Here are recommendations on when to use:

You want to get explicit symbolic representations of the different values a variable could take that would satisfy the equation.
You want to substitute those explicit solution values into other equations or expressions involving the same variable using
subs()

You want to represent the solutions in a mathematically precise way, using mathematical sets.
You want a representation of all the solutions, including if there are infinitely many.
You want a consistent input interface.
You want to limit the domain of the solutions to any arbitrary set.
You do not need to programmatically extract solutions from the solution set: solution sets cannot necessarily be interrogated programmatically.
Guidance¶
Refer to Include the Variable to be Solved for in the Function Call and Ensure Consistent Formatting From solve().
Solve an Equation Algebraically¶
You can solve an equation in several ways. The examples below demonstrate using
both solve()
and solveset()
where applicable. You can choose the
function best suited to your equation.
Make Your Equation Into an Expression That Equals Zero¶
Use the fact that any expression not in an Eq
(equation) is automatically
assumed to equal zero (0) by the solving functions. You can rearrange the
equation \(x^2 = y\) to \(x^2  y = 0\), and solve that expression. This approach is
convenient if you are interactively solving an expression which already equals
zero, or an equation that you do not mind rearranging to \(expression = 0\).
>>> from sympy import solve, solveset
>>> from sympy.abc import x, y
>>> solve(x**2  y, x, dict=True)
[{x: sqrt(y)}, {x: sqrt(y)}]
>>> solveset(x**2  y, x)
{sqrt(y), sqrt(y)}
Put Your Equation Into Eq
Form¶
Put your equation into Eq
form, then solve the Eq
. This approach is
convenient if you are interactively solving an equation which you already have
in the form of an equation, or which you think of as an equality. It also helps
to prevent sign errors when subtracting one side from the other.
>>> from sympy import Eq, solve, solveset
>>> from sympy.abc import x, y
>>> eqn = Eq(x**2, y)
>>> eqn
Eq(x**2, y)
>>> solutions = solve(eqn, x, dict=True)
>>> print(solutions)
[{x: sqrt(y)}, {x: sqrt(y)}]
>>> solutions_set = solveset(eqn, x)
>>> print(solutions_set)
{sqrt(y), sqrt(y)}
>>> for solution_set in solutions_set:
... print(solution_set)
sqrt(y)
sqrt(y)
Restrict the Domain of Solutions¶
By default, SymPy will return solutions in the complex domain, which also includes purely real and imaginary values. Here, the first two solutions are real, and the last two are imaginary:
>>> from sympy import Symbol, solve, solveset
>>> x = Symbol('x')
>>> solve(x**4  256, x, dict=True)
[{x: 4}, {x: 4}, {x: 4*I}, {x: 4*I}]
>>> solveset(x**4  256, x)
{4, 4, 4*I, 4*I}
To restrict returned solutions to real numbers, or another domain or range, the different solving functions use different methods.
For solve()
, place an assumption on the symbol to be solved for, \(x\)
>>> from sympy import Symbol, solve
>>> x = Symbol('x', real=True)
>>> solve(x**4  256, x, dict=True)
[{x: 4}, {x: 4}]
or restrict the solutions with standard Python techniques for filtering a list such as a list comprehension:
>>> from sympy import Or, Symbol, solve
>>> x = Symbol('x', real=True)
>>> expr = (x4)*(x3)*(x2)*(x1)
>>> solution = solve(expr, x)
>>> print(solution)
[1, 2, 3, 4]
>>> solution_outside_2_3 = [v for v in solution if (v.is_real and Or(v<2,v>3))]
>>> print(solution_outside_2_3)
[1, 4]
For solveset()
, limit the output domain in the function call by setting
a domain
>>> from sympy import S, solveset
>>> from sympy.abc import x
>>> solveset(x**4  256, x, domain=S.Reals)
{4, 4}
or by restricting returned solutions to any arbitrary set, including an interval:
>>> from sympy import Interval, pi, sin, solveset
>>> from sympy.abc import x
>>> solveset(sin(x), x, Interval(pi, pi))
{0, pi, pi}
and if you restrict the solutions to a domain in which there are no solutions,
solveset()
will return the empty set,
EmptySet:
>>> from sympy import solveset, S
>>> from sympy.abc import x
>>> solveset(x**2 + 1, x, domain=S.Reals)
EmptySet
Explicitly Represent Infinite Sets of Possible Solutions¶
solveset()
can represent infinite sets of possible
solutions and express them in standard mathematical notation, for
example \(\sin(x) = 0\) for \(x = n * \pi\) for every integer value of \(n\):
>>> from sympy import pprint, sin, solveset
>>> from sympy.abc import x
>>> solution = solveset(sin(x), x)
>>> pprint(solution)
{2*n*pi  n in Integers} U {2*n*pi + pi  n in Integers}
However, solve()
will return only a finite number of solutions:
>>> from sympy import sin, solve
>>> from sympy.calculus.util import periodicity
>>> from sympy.abc import x
>>> f = sin(x)
>>> solve(f, x)
[0, pi]
>>> periodicity(f, x)
2*pi
solve()
tries to return just enough solutions so that all (infinitely
many) solutions can generated from the returned solutions by adding integer
multiples of the periodicity()
of the equation, here \(2\pi\).
Use the Solution Result¶
Substitute Solutions From solve()
Into an Expression¶
You can substitute solutions from solve()
into an expression.
A common use case is finding the critical points and values for a function \(f\).
At the critical points, the Derivative
equals zero (or is undefined).
You can then obtain the function values at those critical points by substituting
the critical points back into the function using
subs()
. You can also tell if the critical point is
a maxima or minima by substituting the values into the expression for the second
derivative: a negative value indicates a maximum, and a positive value indicates
a minimum.
>>> from sympy.abc import x
>>> from sympy import solve, diff
>>> f = x**3 + x**2  x
>>> derivative = diff(f, x)
>>> critical_points = solve(derivative, x, dict=True)
>>> print(critical_points)
[{x: 1}, {x: 1/3}]
>>> point1, point2 = critical_points
>>> print(f.subs(point1))
1
>>> print(f.subs(point2))
5/27
>>> curvature = diff(f, x, 2)
>>> print(curvature.subs(point1))
4
>>> print(curvature.subs(point2))
4
solveset()
Solution Sets Cannot Necessarily Be Interrogated Programmatically¶
If solveset()
returns a finite set (class FiniteSet
), you can
iterate through the solutions:
>>> from sympy import solveset
>>> from sympy.abc import x, y
>>> solution_set = solveset(x**2  y, x)
>>> print(solution_set)
{sqrt(y), sqrt(y)}
>>> solution_list = list(solution_set)
>>> print(solution_list)
[sqrt(y), sqrt(y)]
However, for more complex results, it may not be possible to list the solutions:
>>> from sympy import S, solveset, symbols
>>> x, y = symbols('x, y')
>>> solution_set = solveset(x**2  y, x, domain=S.Reals)
>>> print(solution_set)
Intersection({sqrt(y), sqrt(y)}, Reals)
>>> list(solution_set)
Traceback (most recent call last):
...
TypeError: The computation had not completed because of the undecidable set
membership is found in every candidates.
In this case, it is because, if \(y\) is negative, its square root would be imaginary rather than real and therefore outside the declared domain of the solution set. By declaring \(y\) to be real and positive, SymPy can determine that its square root is real, and thus resolve the intersection between the solutions and the set of real numbers:
>>> from sympy import S, Symbol, solveset
>>> x = Symbol('x')
>>> y = Symbol('y', real=True, positive=True)
>>> solution_set = solveset(x**2  y, x, domain=S.Reals)
>>> print(solution_set)
{sqrt(y), sqrt(y)}
>>> list(solution_set)
[sqrt(y), sqrt(y)]
Alternatively, you can extract the sets from the solution set using args
, then create a list from the set containing the
symbolic solutions:
>>> from sympy import S, solveset, symbols
>>> x, y = symbols('x, y')
>>> solution_set = solveset(x**2  y, x, domain=S.Reals)
>>> print(solution_set)
Intersection({sqrt(y), sqrt(y)}, Reals)
>>> solution_set_args = solution_set.args
>>> print(solution_set.args)
(Reals, {sqrt(y), sqrt(y)})
>>> list(solution_set_args[1])
[sqrt(y), sqrt(y)]
Options That Can Speed up solve()
¶
Refer to solving guidance.
Not All Equations Can Be Solved¶
Equations With No ClosedForm Solution¶
Some equations have no closedform solution, in which case SymPy may return an empty set or give an error. For example, the following transcendental equation has no closedform solution:
>>> from sympy import cos, solve
>>> from sympy.abc import x
>>> solve(cos(x)  x, x, dict=True)
Traceback (most recent call last):
...
NotImplementedError: multiple generators [x, cos(x)]
No algorithms are implemented to solve equation x + cos(x)
Equations Which Have a ClosedForm Solution, and SymPy Cannot Solve¶
It is also possible that there is an algebraic solution to your equation, and
SymPy has not implemented an appropriate algorithm. If that happens, or SymPy
returns an empty set or list when there is a mathematical solution (indicating a
bug in SymPy), please post it on the mailing
list, or open an issue on SymPy’s GitHub
page. Until the issue is resolved, you
can solve your equation numerically
instead.
Report a Bug¶
If you find a bug with a solving function, please post the problem on the SymPy mailing list. Until the issue is resolved, you can use a different method listed in Alternatives to consider.