6 Special ode’s and their solutions

6.1 Introduction and table lookup

These are ode’s whose solution is in terms of special functions. Will update as I find more. Most of the special functions come up from working out the solution in series of a second order ode which has regular singular point at expansion point. These are the more interesting odes whose solution is in terms of special functions.

Given an ode of the form \(A y^{\prime \prime }+ B y^{\prime }+ C y=0\), we first look at \(\frac {B}{A}\) and \(\frac {C}{A}\). If both of these are analytic at the expansion point \(x_{0}\), then \(x_{0}\) is called an ordinary point. If at least one of these is not analytic, then the point \(x_{0}\) is called singular point. Normally \(A,B,C\) are polynomials in \(x\).

If \(x_{0}\) is not analytic based on the above check, then we now do an additional test and look at \((x-x_{0}) \frac {B}{A}\) and also at \((x-x_{0})^{2} \frac {C}{A}\) instead, and if now both of these are analytic at \(x_{0}\), then the point \(x_{0}\) is called a regular singular point (which means removable singularity).

If at least one of the above tests fail, then \(x_{0}\) is not analytic, and it is called irregular singular point or essential singularity.

Only when the expansion point \(x_{0}\) is ordinary or regular singular point can we do series solution for the ode. If it is irregular singular point then asymptotic expansion is needed.

For ordinary point, standard power series \(y(x)=\sum _{n=0}^{\infty } a_{n} x^{n}\) is used. For regular singular point, Frobenius method \(y(x)=\sum _{n=0}^{\infty } a_{n} x^{n+r}\) is used.

Table 1: Special second order differential equations lookup table

#	Name	ode	ordinary and singular points
1	Airy	\( y'' + A y = 0\)	All points are ordinary
2	Bessel	\(x^2 y'' + x y' + (x^2 - n^2) y = 0 \)	All points are ordinary except \(x=0\) is regular singular point
3	Chebyshev	\((1-x^2) y'' - x y' +n^2 y =0\)	All points are ordinary except \(x=\pm 1\) are regular singular points
4	Hermite	\( y'' -2 x y +2 n y=0 \)	All points are ordinary
5	Gauss	\( x(1-x) y'' + (c - (a+b+1)x ) y' - a b y=0 \)	All points are ordinary except \(x=0,1\) are regular singular points
6	Laguerre	\( x y'' + (1-x) y' + n y=0 \)	All points are ordinary except \(x=0\) is regular singular point
7	Legendre	\( (1-x^2) y'' -2 x y' + n(n+1) y=0 \)	All points are ordinary except \(x=\pm 1\) are regular singular points

6.2 Airy  \(y^{\prime \prime }+axy=0\)

solution is

\[ y\left ( x\right ) =c_{1}\operatorname {AiryAi}\left ( -a^{\frac {1}{3}}x\right ) +c_{2}\operatorname {AiryBi}\left ( -a^{\frac {1}{3}}x\right ) \]

6.3 Chebyshev \(\left ( 1-x^{2}\right ) y^{\prime \prime }-xy^{\prime }+n^{2}y=0\)

For

\[ \left ( 1-x^{2}\right ) y^{\prime \prime }-xy^{\prime }+n^{2}y=0 \]

Singular points at \(x=1,-1\) and \(\infty \). Solution valid for \(\left \vert x\right \vert <1\). Maple gives solution

\[ y\left ( x\right ) =c_{1}\frac {1}{\left ( x+\sqrt {x^{2}-1}\right ) ^{n}}+c_{2}\left ( x+\sqrt {x^{2}-1}\right ) ^{n}\]

For

\[ \left ( 1-x^{2}\right ) y^{\prime \prime }-axy^{\prime }+n^{2}y=0 \]

Maple gives solution

\begin{multline*} y\left ( x\right ) =c_{1}\left ( x^{2}-1\right ) ^{\frac {1}{2}-\frac {a}{4}}\operatorname {LegendreP}\left ( \frac {\sqrt {a^{2}+4n^{2}-2a+1}}{2}-\frac {1}{2},-1+\frac {a}{2},x\right ) \\ +c_{2}\left ( x^{2}-1\right ) ^{\frac {1}{2}-\frac {a}{4}}\operatorname {LegendreQ}\left ( \frac {\sqrt {a^{2}+4n^{2}-2a+1}}{2}-\frac {1}{2},-1+\frac {a}{2},x\right ) \end{multline*}

If \(n\) positive integer, then solution in series gives polynomial solution of degree \(n\). Called Chebyshev polynomials.

6.4 Hermite \(y^{\prime \prime }-2xy^{\prime }+2ny=0\)

Converges for all \(x\). If \(n\) is positive integer, one series terminates. Series solution in terms of Hermite polynomials.

Maple gives solution

\[ y\left ( x\right ) =c_{1}x\operatorname {KummerM}\left ( \frac {1}{2}-\frac {n}{2},\frac {3}{2},x^{2}\right ) +c_{2}x\operatorname {KummerU}\left ( \frac {1}{2}-\frac {n}{2},\frac {3}{2},x^{2}\right ) \]

6.5 Legendre \(\left ( 1-x^{2}\right ) y^{\prime \prime }-2xy^{\prime }+n\left ( n+1\right ) y=0\)

Series solution in terms of Legendre functions. When \(n\) is positive integer, one series terminates (i.e. becomes polynomial).

Maple gives solution

\[ y\left ( x\right ) =c_{1}\operatorname {LegendreP}\left ( n,x\right ) +c_{2}\operatorname {LegendreQ}\left ( n,x\right ) \]

If the ode is given in form

\[ \sin \left ( \theta \right ) P^{\prime \prime }\left ( \theta \right ) +\cos \left ( \theta \right ) P^{\prime }\left ( \theta \right ) +n\sin \left ( \theta \right ) P\left ( \theta \right ) =0 \]

Then using \(x=\cos \theta \) transforms it to the earlier more familiar form. Maple gives this as solution

\[ P\left ( \theta \right ) =c_{1}\operatorname {LegendreP}\left ( \frac {\sqrt {4n+1}}{2}-\frac {1}{2},\cos \theta \right ) +c_{2}\operatorname {LegendreQ}\left ( \frac {\sqrt {4n+1}}{2}-\frac {1}{2},\cos \theta \right ) \]

6.6 Bessel \(x^{2}y^{\prime \prime }+xy^{\prime }+\left ( x^{2}-n^{2}\right ) y=0\)

\(x=0\,\) is regular singular point. Solution in terms of Bessel functions

\[ y\left ( x\right ) =c_{1}\operatorname {BesselJ}\left ( n,x\right ) +c_{2}\operatorname {BesselY}\left ( n,x\right ) \]

6.7 Reduced Riccati \(y^{\prime }=ax^{n}+by^{2}\)

For the special case of \(n=-2\) the solution is

\[ y\left ( x\right ) =\frac {\lambda }{x}-\frac {x^{2b\lambda }}{\frac {bx}{2b\lambda +1}x^{2b\lambda }+c_{1}}\]

Where in the above \(\lambda \) is a root of \(b\lambda ^{2}+\lambda +a=0\).

For \(n\neq -2\)

\begin{align*} w & =\sqrt {x}\left \{ \begin {array} [c]{cc}c_{1}\operatorname {BesselJ}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab}x^{k}\right ) +c_{2}\operatorname {BesselY}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {ab}x^{k}\right ) & ab>0\\ c_{1}\operatorname {BesselI}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {-ab}x^{k}\right ) +c_{2}\operatorname {BesselK}\left ( \frac {1}{2k},\frac {1}{k}\sqrt {-ab}x^{k}\right ) & ab<0 \end {array} \right . \\ y & =-\frac {1}{b}\frac {w^{\prime }}{w}\\ k & =1+\frac {n}{2}\end{align*}

6.8 Gauss Hypergeometric ode \(x\left ( 1-x\right ) y^{\prime \prime }+\left ( c-\left ( a+b+1\right ) x\right ) y^{\prime }-aby=0\)

Solution is for \(\left \vert x\right \vert <1\) is in terms of hypergeom function. Has 3 regular singular points, \(x=0,x=1,x=\infty \).

Maple gives this solution

\[ y\left ( x\right ) =c_{1}\operatorname {hypergeom}\left ( \left [ a,b\right ] ,\left [ c\right ] ,x\right ) +c_{2}x^{1-c}\operatorname {hypergeom}\left ( \left [ 1+a-c,1+b-c\right ] ,\left [ 2-c\right ] ,x\right ) \]

And Mathematica gives

\[ y\left ( x\right ) =c_{1}\operatorname {HypergeometricF1}\left ( a,b,c,x\right ) +\left ( -1\right ) ^{1-c}x^{1-c}c_{2}\operatorname {HypergeometricF1}\left ( 1+a-c,1+b-c,2-c,x\right ) \]