Explore multiple solutions, constraints and exact integers.

Multiple solutions without starting guesses

Many nonlinear systems have more than one valid real solution. The engine searches for distinct roots automatically using supported mathematical expressions, variable domains and settings. It reports numerical residuals and diagnostic information that help a user evaluate the discovered results.

var x,y
x^2 + y^2 = 25
x*y = 12

This system has four real solutions: (3,4), (4,3), (−3,−4) and (−4,−3). The keyword var is used to control the order in which variables are printed in solution output; the variables already occur in the equations. This display directive is not the same as the integration-variable setting in quad.

Integer, mixed-integer and number-theory problems

Use declarations such as int x,y and pos int x,y for supported discrete variables. Declare bounds to exclude irrelevant branches; supported exact-integer routes can handle large values without rounding every integer to a floating-point approximation.

pos int x,y
x^2 - 2*y^2 = 1
bound x = [1,1000]
bound y = [1,1000]

Pell-type equations illustrate why integer declarations matter. Other supported number-theory families include linear Diophantine equations, quadratic Diophantine forms and weighted sums of squares. Inspect the engine’s completion indicators for exact enumeration; they are distinct from ordinary numerical-search status.

Practical formulation guidance