An ordinary differential equation whose only singular points, including the point at infinity, are regular singular points is called a Fuchsian ordinary differential equation. Examples for second order differential equations In this case the equation above is reduced to: One distinguishes the following cases: Point is an ordinary point when functions and are analytic at . Point is a regular singular point if has a pole up to order 1 at and has a pole of order up to 2 at . Otherwise point is an irregular singular point. We can check whether there is an irregular singular point at infinity by using the substitution and the relations: We can thus transform the equation to an equation in , and check what happens at . If and are quotients of polynomials, then there will be an irregular singular point at infinite x unless the polynomial in the denominator of is of degree at least one more than the degree of its numerator and the denominator of is of degree at least two more than the degree of its numerator. Listed below are several examples from ordinary differential equations from mathematical physics that have singular points and known solutions. Bessel differential equation This is an ordinary differential equation of second order. It is found in the solution to Laplace's equation in cylindrical coordinates: for an arbitrary real or complex number (the order of the Bessel function). The most common and important special case is where is an integer . Dividing this equation by x2 gives: In this case has a pole of first order at . When , has a pole of second order at . Thus this equation has a regular singularity at 0. To see what happens when one has to use a Möbius transformation, for example . After performing the algebra: Now at has a pole of first order, but has a pole of fourth order. Thus, this equation has an irregular singularity at corresponding to x at ∞. Legendre differential equation This is an ordinary differential equation of second order. It is found in the solution of Laplace's equation in spherical coordinates: Opening the square bracket gives: And dividing by : This differential equation has regular singular points at ±1 and ∞. Hermite differential equation One encounters this ordinary second order differential equation in solving the one-dimensional time independent Schrödinger equation for a harmonic oscillator. In this