sympy

Use SymPy's solveset for algebraic equations, linsolve for linear systems, nonlinsolve for nonlinear systems, and dsolve for differential equations. Define symbols first with symbols('x'), then pass the equation and variable to the appropriate solver.

Use diff(expr, x) for derivatives and integrate(expr, x) for indefinite integrals, or integrate(expr, (x, a, b)) for definite ones. SymPy returns exact symbolic results like x**3/3 rather than decimal approximations.

SymPy provides exact symbolic results for algebra and calculus, while NumPy handles fast numerical arrays. A common pattern is deriving expressions symbolically in SymPy, then using lambdify to convert them into NumPy functions for high-performance evaluation.

Writing 0.5*x creates a floating-point approximation instead of an exact value. Use Rational(1, 2) or S(1)/2 for exact rational arithmetic so simplification and solving remain precise.

Repeatedly calling subs() and evalf() is slow because each call runs symbolic machinery. Convert the expression once with lambdify(x, expr, 'numpy') to get a compiled numerical function that evaluates arrays quickly.

Yes, sympy.utilities.codegen.codegen produces C and Fortran source from symbolic expressions, and autowrap compiles them into callable Python functions. The latex() function also generates LaTeX output for documentation.