Counting roots of polynomials. CountRoots accepts polynomials with Gaussian rational coefficients. The root count includes multiplicities. This gives the number of real roots ...

Structural operations on polynomials. Here is a polynomial in one variable. Expand expands out products and powers, writing the polynomial as a simple sum of terms.

NRoots[lhs == rhs, var] yields a disjunction of equations which represent numerical approximations to the roots of a polynomial equation.

Orthogonal polynomials. Legendre polynomials LegendreP[n,x] arise in studies of systems with three-dimensional spherical symmetry. They satisfy the differential equation ...

These "How tos" give step-by-step instructions for common tasks related to solving differential equations in Mathematica .

It may happen that a given ODE is not linear in y(x) but can be viewed as a linear ODE in x(y). In this case, it is said to be an inverse linear ODE. This is an inverse ...

An Euler equation has the general form Euler equations can be solved by transforming them to equations with constant coefficients. This is an example of an Euler equation.

Mathematica 's differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms ...

Mathematica 's functions for solving differential equations can be applied to many different classes of differential equations, including ordinary differential equations ...

While differential equations have three basic types—ordinary (ODEs), partial (PDEs), or differential-algebraic (DAEs), they can be further described by attributes such as ...