# Solvers¶

```
>>> from sympy import *
>>> x, y, z = symbols('x y z')
>>> init_printing(use_unicode=True)
```

## A Note about Equations¶

Recall from the gotchas section of this
tutorial that symbolic equations in SymPy are not represented by `=`

or
`==`

, but by `Eq`

.

```
>>> Eq(x, y)
x = y
```

However, there is an even easier way. In SymPy, any expression not in an
`Eq`

is automatically assumed to equal 0 by the solving functions. Since \(a
= b\) if and only if \(a - b = 0\), this means that instead of using `x == y`

,
you can just use `x - y`

. For example

```
>>> solveset(Eq(x**2, 1), x)
{-1, 1}
>>> solveset(Eq(x**2 - 1, 0), x)
{-1, 1}
>>> solveset(x**2 - 1, x)
{-1, 1}
```

This is particularly useful if the equation you wish to solve is already equal
to 0. Instead of typing `solveset(Eq(expr, 0), x)`

, you can just use
`solveset(expr, x)`

.

## Solving Equations Algebraically¶

The main function for solving algebraic equations is `solveset`

.
The syntax for `solveset`

is `solveset(equation, variable=None, domain=S.Complexes)`

Where `equations`

may be in the form of `Eq`

instances or expressions
that are assumed to be equal to zero.

Please note that there is another function called `solve`

which
can also be used to solve equations. The syntax is `solve(equations, variables)`

However, it is recommended to use `solveset`

instead.

When solving a single equation, the output of `solveset`

is a `FiniteSet`

or
an `Interval`

or `ImageSet`

of the solutions.

```
>>> solveset(x**2 - x, x)
{0, 1}
>>> solveset(x - x, x, domain=S.Reals)
ℝ
>>> solveset(sin(x) - 1, x, domain=S.Reals)
⎧ π ⎫
⎨2⋅n⋅π + ─ | n ∊ ℤ⎬
⎩ 2 ⎭
```

If there are no solutions, an `EmptySet`

is returned and if it
is not able to find solutions then a `ConditionSet`

is returned.

```
>>> solveset(exp(x), x) # No solution exists
∅
>>> solveset(cos(x) - x, x) # Not able to find solution
{x | x ∊ ℂ ∧ -x + cos(x) = 0}
```

In the `solveset`

module, the linear system of equations is solved using `linsolve`

.
In future we would be able to use linsolve directly from `solveset`

. Following
is an example of the syntax of `linsolve`

.

List of Equations Form:

>>> linsolve([x + y + z - 1, x + y + 2*z - 3 ], (x, y, z)) {(-y - 1, y, 2)}

Augmented Matrix Form:

>>> linsolve(Matrix(([1, 1, 1, 1], [1, 1, 2, 3])), (x, y, z)) {(-y - 1, y, 2)}

A*x = b Form

>>> M = Matrix(((1, 1, 1, 1), (1, 1, 2, 3))) >>> system = A, b = M[:, :-1], M[:, -1] >>> linsolve(system, x, y, z) {(-y - 1, y, 2)}

Note

The order of solution corresponds the order of given symbols.

In the `solveset`

module, the non linear system of equations is solved using
`nonlinsolve`

. Following are examples of `nonlinsolve`

.

When only real solution is present:

>>> a, b, c, d = symbols('a, b, c, d', real=True) >>> nonlinsolve([a**2 + a, a - b], [a, b]) {(-1, -1), (0, 0)} >>> nonlinsolve([x*y - 1, x - 2], x, y) {(2, 1/2)}

When only complex solution is present:

>>> nonlinsolve([x**2 + 1, y**2 + 1], [x, y]) {(-ⅈ, -ⅈ), (-ⅈ, ⅈ), (ⅈ, -ⅈ), (ⅈ, ⅈ)}

When both real and complex solution is present:

>>> from sympy import sqrt >>> system = [x**2 - 2*y**2 -2, x*y - 2] >>> vars = [x, y] >>> nonlinsolve(system, vars) {(-2, -1), (2, 1), (-√2⋅ⅈ, √2⋅ⅈ), (√2⋅ⅈ, -√2⋅ⅈ)}

>>> n = Dummy('n') >>> system = [exp(x) - sin(y), 1/y - 3] >>> real_soln = (log(sin(S(1)/3)), S(1)/3) >>> img_lamda = Lambda(n, 2*n*I*pi + Mod(log(sin(S(1)/3)), 2*I*pi)) >>> complex_soln = (ImageSet(img_lamda, S.Integers), S(1)/3) >>> soln = FiniteSet(real_soln, complex_soln) >>> nonlinsolve(system, [x, y]) == soln True

4. If non linear system of equations is Positive dimensional system (A system with infinitely many solutions is said to be positive-dimensional):

```
>>> nonlinsolve([x*y, x*y - x], [x, y])
{(0, y)}
```

```
>>> system = [a**2 + a*c, a - b]
>>> nonlinsolve(system, [a, b])
{(0, 0), (-c, -c)}
```

Note

- The order of solution corresponds the order of given symbols.

2. Currently `nonlinsolve`

doesn’t return solution in form of `LambertW`

(if there
is solution present in the form of `LambertW`

).

`solve`

can be used for such cases:

```
>>> solve([x**2 - y**2/exp(x)], [x, y], dict=True)
⎡⎧ ⎛y⎞⎫⎤
⎢⎨x: 2⋅LambertW⎜─⎟⎬⎥
⎣⎩ ⎝2⎠⎭⎦
```

3. Currently `nonlinsolve`

is not properly capable of solving the system of equations
having trigonometric functions.

`solve`

can be used for such cases(not all solution):

```
>>> solve([sin(x + y), cos(x - y)], [x, y])
⎡⎛-3⋅π 3⋅π⎞ ⎛-π π⎞ ⎛π 3⋅π⎞ ⎛3⋅π π⎞⎤
⎢⎜─────, ───⎟, ⎜───, ─⎟, ⎜─, ───⎟, ⎜───, ─⎟⎥
⎣⎝ 4 4 ⎠ ⎝ 4 4⎠ ⎝4 4 ⎠ ⎝ 4 4⎠⎦
```

`solveset`

reports each solution only once. To get the solutions of a
polynomial including multiplicity use `roots`

.

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

The output `{0: 1, 3: 2}`

of `roots`

means that `0`

is a root of
multiplicity 1 and `3`

is a root of multiplicity 2.

Note

Currently `solveset`

is not capable of solving the following types of equations:

- Equations solvable by LambertW (Transcendental equation solver).

`solve`

can be used for such cases:

```
>>> solve(x*exp(x) - 1, x )
[LambertW(1)]
```

## Solving Differential Equations¶

To solve differential equations, use `dsolve`

. First, create an undefined
function by passing `cls=Function`

to the `symbols`

function.

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

`f`

and `g`

are now undefined functions. We can call `f(x)`

, and it
will represent an unknown function.

```
>>> f(x)
f(x)
```

Derivatives of `f(x)`

are unevaluated.

```
>>> f(x).diff(x)
d
──(f(x))
dx
```

(see the Derivatives section for more on derivatives).

To represent the differential equation \(f''(x) - 2f'(x) + f(x) = \sin(x)\), we would thus use

```
>>> diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))
>>> diffeq
2
d d
f(x) - 2⋅──(f(x)) + ───(f(x)) = sin(x)
dx 2
dx
```

To solve the ODE, pass it and the function to solve for to `dsolve`

.

```
>>> dsolve(diffeq, f(x))
x cos(x)
f(x) = (C₁ + C₂⋅x)⋅ℯ + ──────
2
```

`dsolve`

returns an instance of `Eq`

. This is because in general,
solutions to differential equations cannot be solved explicitly for the
function.

```
>>> dsolve(f(x).diff(x)*(1 - sin(f(x))) - 1, f(x))
-x + f(x) + cos(f(x)) = C₁
```

The arbitrary constants in the solutions from dsolve are symbols of the form
`C1`

, `C2`

, `C3`

, and so on.