SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible.

This post only shows some examples for solving equations, derivatives and integrals. For more complete tutorials please check the SymPy Tutorial website.

from sympy import*

x, y, z = symbols("x, y, z", real=True)
init_printing(use_latex="mathjax")

Solve

solve(Eq(x-2,4),x)

$\displaystyle \left[ 6\right]$

If the argument of the solve function does not contain the word “Eq”, thus SymPy will assume that the equation is equal to zero.

solve((x-3)*(x-2),x)

$\displaystyle \left[ 2, \ 3\right]$

solve(5*x**2 + 6*x + 1, x)

$\displaystyle \left[ -1, \ - \frac{1}{5}\right]$

solve([Eq(2*x+4*y, 22), Eq(-2*x+2*y, 2)], [x,y])

$\displaystyle \left{ x : 3, \ y : 4\right}$

solve([2*x+4*y-22, -2*x+2*y-2], [x,y])

$\displaystyle \left{ x : 3, \ y : 4\right}$

Derivatives

To take derivatives, use the diff function.

diff(cos(x), x)

$\displaystyle - \sin{\left(x \right)}$

diff can take multiple derivatives at once. To take multiple derivatives, pass the variable as many times as you wish to differentiate, or pass a number after the variable. For example, both of the following find the third derivative of $x^4$.

diff(x**4, x, x, x)

$\displaystyle 24 x$

diff(x**4, x, 3)

$\displaystyle 24 x$

You can also take derivatives with respect to many variables at once. Just pass each derivative in order, using the same syntax as for single variable derivatives. For example, each of the following will compute

expr = exp(x*y*z)
diff(expr, x, y, y, z, z, z, z)

$\displaystyle x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$

diff(expr, x, y, 2, z, 4)

$\displaystyle x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$

diff(expr, x, y, y, z, 4)

$\displaystyle x^{3} y^{2} \left(x^{3} y^{3} z^{3} + 14 x^{2} y^{2} z^{2} + 52 x y z + 48\right) e^{x y z}$

To print the derivative, use the Derivative class.

Derivative(expr, x, y, y, z, 4)

$\displaystyle \frac{\partial^{7}}{\partial z^{4}\partial y^{2}\partial x} e^{x y z}$

Integrals

$\displaystyle \int (x-2)\ dx$

To compute this integral, use the integrate function.

f = x-2

integrate(f)

$\displaystyle \frac{x^{2}}{2} - 2 x$

Use the command Integral to print an integral.

Integral(f,x)

$\displaystyle \int \left(x - 2\right)\, dx$

expr = Integral(log(x)**2, x)
expr

$\displaystyle x \log{\left(x \right)}^{2} - 2 x \log{\left(x \right)} + 2 x$

To later evaluate the integral, call doit.

expr.doit()

$\displaystyle x \log{\left(x \right)}^{2} - 2 x \log{\left(x \right)} + 2 x$

To compute a definite integral, pass the argument (integration_variable, lower_limit, upper_limit). For example, to compute

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

$\displaystyle 1$

As with indefinite integrals, you can pass multiple limit tuples to perform a multiple integral. For example, to compute

integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))

$\displaystyle \pi$

References

1. SymPy 1.10.1 documentation » SymPy Tutorial