Consider again the eigenvalue equation, Equation (). Both equations reduce to the single linear equation formula 187. They arise in solving matrix equations such as the Sylvester equation. These equations are merely obtained by making formula 69 in the step two equation. Algebraic numbers are those that are a solution to a polynomial equation with integer coefficients. The diffusion equation is a parabolic partial differential equation. The solutions of a homogeneous linear differential equation form a vector space. Let the roots of the standard quadratic equation be and . More generally, every element in satisfies the polynomial equation . The heat equation is a partial differential equation. This is an ordinary differential equation for the coordinates. The same equation (modulo constants) results. This is an ordinary differential equation (ODE). Galois theory concerns transformations of number fields that permute the roots of an equation. A non-linear ordinary differential equation for the normal quantile, "w"("p"), may be given.