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Symbols

To make symbolic variables in SymPy you have to declare the variable explicitly:

>>> from sympy import *

>>> x = Symbol('x')
>>> y = Symbol('y')

You may then manipulate them as you want:

>>> x + y + x - y
2*x

>>> (x + y)**2
(x + y)**2

>>> ((x + y)**2).expand()
x**2 + 2*x*y + y**2

You can also substitute variables for numbers or other symbolic variables with subs(var, substitution).

>>> ((x + y)**2).subs(x, 1)
(y + 1)**2

>>> ((x + y)**2).subs(x, y)
4*y**2

Some regular constants are already included in SymPy as symbols, like e, pi and infinite. evalf() evaluates the symbols as floation-point numbers.

>>> pi**2
pi**2

>>> pi.evalf()
3.141592653589793238462643383

<<< E**2
exp(2)

>>> oo > 99999
True

>>> oo + 1
oo

Differentiation

You can differentiate any SymPy expression using diff(func, var). Higher derivatives can be solved using diff(func, var, n).

>>> from sympy import *
>>> x = Symbol('x')

>>> diff(sin(x), x)
cos(x)

>>> diff(sin(2*x), x, 1)
2*cos(2*x)

>>> diff(sin(2*x), x, 2)
-4*sin(2*x)

Integration

SymPy has support for both indefinite and definite integration.

>>> from sympy import *
>>> x, y = symbols('x y')

Indefinite integration of some elementary functions:

>>> integrate(6*x**5, x)
x**6

>>> integrate(log(x), x)
x*log(x) - x

>>> integrate(2*x + sinh(x), x)
x**2 + cosh(x)	

Definite integration:

>>> integrate(x**3, (x, -1, 1))
0

>>> integrate(sin(x), (x, 0, pi/2))
1

Some special integrals:

>>> integrate(exp(-x), (x, 0, oo))
1

>>> integrate(log(x), (x, 0, 1))
-1

Algebraic equations

SymPy is able to solve algebraic equations with one or several variables. solve(equation, variable) takes as first argument an equation that is supposed to be equaled to 0, and as second argument the variable to solve for. In case of multiple equations and variables, solve returns a dictonary containing the results. Note that SymPy also handles complex numbers using the symbol I.

>>> from sympy import *
>>> x = Symbol('x')

>>> solve(x**2 - 1, x)
[-1, 1]

>>> solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y])
{x: -3, y: 1}

>>> solve(Eq(x**4, 1), x)
[-1, 1, -I, I]

nsolve(function,[variables], x0) is a useful tool to solve nonlinear equation system numerically, where x0 is a starting vector close to the solution. If there is only one variable, the second argument may be left out.

>>> nsolve(sin(x), x, 2)
3.14159265358979

>>> nsolve(sin(x), 2)
3.14159265358979

>>> nsolve(sin(x**2)/(pi - x), x, pi.evalf()/2)
1.77245385090552