Special ode’s and their solutions

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 second order ode which has regular singular point at expansion point. These are the more interesting odes which will generate these special functions.

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

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

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

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

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

\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.

3.3 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.

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

3.4 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).

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

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

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

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

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

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

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

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

\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*}

3.7 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, \(x=0,x=1,x=\infty \).

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

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