§32.9 Other Elementary Solutions

Elementary nonrational solutions of ${\text{P}}_{\text{III}}$ are

32.9.1		$$w(z;\mu ,0,0,-\mu {\kappa }^3)=\kappa z^{1/3},$$
ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E1 Encodings: TeX, pMML, png See also: Annotations for §32.9(i), §32.9 and Ch.32

32.9.2		$$w(z;0,-2\kappa ,0,4\kappa \mu -{\lambda }^2)=z(\kappa {(\ln z)}^2+\lambda \ln z+\mu ),$$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $z$: real, $\kappa$: constant, $\lambda$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E2 Encodings: TeX, pMML, png See also: Annotations for §32.9(i), §32.9 and Ch.32

32.9.3		$$w(z;-{\nu }^2\lambda ,0,{\nu }^2({\lambda }^2-4\kappa \mu ),0)=\frac{z^{\nu -1}}{\kappa z^{2\nu }+\lambda z^{\nu }+\mu },$$
ⓘ Symbols: $z$: real, $\kappa$: constant, $\lambda$: constant, $\mu$: constant and $\nu$: constant Permalink: http://dlmf.nist.gov/32.9.E3 Encodings: TeX, pMML, png See also: Annotations for §32.9(i), §32.9 and Ch.32

with $\kappa$, $\lambda$, $\mu$, and $\nu$ arbitrary constants.

In the case $\gamma =0$ and $\alpha \delta \ne 0$ we assume, as in §32.2(ii), $\alpha =1$ and $\delta =-1$. Then ${\text{P}}_{\text{III}}$ has algebraic solutions iff

32.9.4		$$\beta =2n,$$
ⓘ Symbols: $n$: integer and $\beta$: arbitrary constant Permalink: http://dlmf.nist.gov/32.9.E4 Encodings: TeX, pMML, png See also: Annotations for §32.9(i), §32.9 and Ch.32

with $n\in \mathbb{Z}$. These are rational solutions in $\zeta =z^{1/3}$ of the form

32.9.5		$$w(z)=P_{n^2+1}(\zeta )/Q_{n^2}(\zeta ),$$
ⓘ Symbols: $n$: integer, $z$: real, $P_{n^2+1}(\zeta )$: polynomials and $Q_{n^2}(\zeta )$: polynomials Permalink: http://dlmf.nist.gov/32.9.E5 Encodings: TeX, pMML, png See also: Annotations for §32.9(i), §32.9 and Ch.32

where $P_{n^2+1}(\zeta )$ and $Q_{n^2}(\zeta )$ are polynomials of degrees $n^2+1$ and $n^2$, respectively, with no common zeros. For examples and plots see Clarkson (2003a) and Milne et al. (1997). Similar results hold when $\delta =0$ and $\beta \gamma \ne 0$.

${\text{P}}_{\text{III}}$ with $\beta =\delta =0$ has a first integral

32.9.6		$$z^2{(w^{\prime })}^2+2zw{w}^{\prime }=(C+2\alpha zw+\gamma z^2{w}^2)w^2,$$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\gamma$: arbitrary constant and $C$: constant Permalink: http://dlmf.nist.gov/32.9.E6 Encodings: TeX, pMML, png See also: Annotations for §32.9(i), §32.9 and Ch.32

with $C$ an arbitrary constant, which is solvable by quadrature. A similar result holds when $\alpha =\gamma =0$. ${\text{P}}_{\text{III}}$ with $\alpha =\beta =\gamma =\delta =0$, has the general solution $w(z)=C{z}^{\mu }$, with $C$ and $\mu$ arbitrary constants.

Elementary nonrational solutions of ${\text{P}}_{\text{V}}$ are

32.9.7		$$w(z;\mu ,-\frac{1}{8},-\mu {\kappa }^2,0)=1+\kappa z^{1/2},$$
ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E7 Encodings: TeX, pMML, png See also: Annotations for §32.9(ii), §32.9 and Ch.32

32.9.8		$$w(z;0,0,\mu ,-\frac{1}{2}{\mu }^2)=\kappa \exp \left(\mu z\right),$$
ⓘ Symbols: $\exp z$: exponential function, $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E8 Encodings: TeX, pMML, png See also: Annotations for §32.9(ii), §32.9 and Ch.32

with $\kappa$ and $\mu$ arbitrary constants.

${\text{P}}_{\text{V}}$, with $\delta =0$, has algebraic solutions if either

32.9.9		$$(\alpha ,\beta ,\gamma )=(\frac{1}{2}{\mu }^2,-\frac{1}{8}{(2n-1)}^2,-1),$$
ⓘ Symbols: $n$: integer, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E9 Encodings: TeX, pMML, png See also: Annotations for §32.9(ii), §32.9 and Ch.32

or

32.9.10		$$(\alpha ,\beta ,\gamma )=(\frac{1}{8}{(2n-1)}^2,-\frac{1}{2}{\mu }^2,1),$$
ⓘ Symbols: $n$: integer, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E10 Encodings: TeX, pMML, png See also: Annotations for §32.9(ii), §32.9 and Ch.32

with $n\in \mathbb{Z}$ and $\mu$ arbitrary. These are rational solutions in $\zeta =z^{1/2}$ of the form

32.9.11		$$w(z)=P_{n^2-n+1}(\zeta )/Q_{n^2-n}(\zeta ),$$
ⓘ Symbols: $n$: integer, $z$: real, $P_{n^2+1}(\zeta )$: polynomials and $Q_{n^2}(\zeta )$: polynomials Permalink: http://dlmf.nist.gov/32.9.E11 Encodings: TeX, pMML, png See also: Annotations for §32.9(ii), §32.9 and Ch.32

where $P_{n^2-n+1}(\zeta )$ and $Q_{n^2-n}(\zeta )$ are polynomials of degrees $n^2-n+1$ and $n^2-n$, respectively, with no common zeros.

${\text{P}}_{\text{V}}$, with $\gamma =\delta =0$, has a first integral

32.9.12		$$z^2{(w^{\prime })}^2={(w-1)}^2(2\alpha w^2+Cw-2\beta ),$$
ⓘ Symbols: $z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant and $C$: constant Permalink: http://dlmf.nist.gov/32.9.E12 Encodings: TeX, pMML, png See also: Annotations for §32.9(ii), §32.9 and Ch.32

with $C$ an arbitrary constant, which is solvable by quadrature. For examples and plots see Clarkson (2005). ${\text{P}}_{\text{V}}$, with $\alpha =\beta =0$ and ${\gamma }^2+2\delta =0$, has solutions $w(z)=C\exp \left(\pm \sqrt{-2\delta }z\right)$, with $C$ an arbitrary constant.

An elementary algebraic solution of ${\text{P}}_{\text{VI}}$ is

32.9.13		$$w(z;\frac{1}{2}{\kappa }^2,-\frac{1}{2}{\kappa }^2,\frac{1}{2}{\mu }^2,\frac{1}{2}(1-{\mu }^2))=z^{1/2},$$
ⓘ Symbols: $z$: real, $\kappa$: constant and $\mu$: constant Permalink: http://dlmf.nist.gov/32.9.E13 Encodings: TeX, pMML, png See also: Annotations for §32.9(iii), §32.9 and Ch.32

with $\kappa$ and $\mu$ arbitrary constants.

Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of ${\text{P}}_{\text{VI}}$ with $\beta =\gamma =0$, $\delta =\frac{1}{2}$. For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b).