Table 1 Families of \({\Pi }({\phi })\) and \(\bigg (\frac{{\Pi }'({\phi })}{{\Pi }({\phi })}\bigg )\), wherein \({\eta } ={\rho }^2-4{\varrho } {\varpi }\) and \({\Psi }=\cosh \left( \frac{1}{4}\,\sqrt{\eta }{\phi } \right) \sinh \left( \frac{1}{4}\,\sqrt{\eta }{\phi } \right)\).
S. No. | Family | Condition(s) | \({\Pi }({\phi })\) | \(\bigg (\frac{{\Pi }'({\phi })}{{\Pi }({\phi })}\bigg )\) |
|---|---|---|---|---|
1 | Trigonometric Solutions | \({\eta }<0, \quad {\varrho }\ne 0\) | \(-{\frac{{\rho }}{2{\varrho }}}+{\frac{\sqrt{-{\eta }}\tan \left( \frac{1}{2}\,\sqrt{-{\eta }}{\phi } \right) }{2{\varrho }}}\), | \(-\frac{1}{2}\,{\frac{{\eta }\, \left( 1+ \left( \tan \left( \frac{1}{2}\,\sqrt{-{\eta }}{\phi } \right) \right) ^{2} \right) }{-{\rho }+\sqrt{-{\eta }}\tan \left( \frac{1}{2}\,\sqrt{-{\eta }}{\phi } \right) }}\), |
\(-{\frac{{\rho }}{2{\varrho }}}-{\frac{\sqrt{-{\eta }}\cot \left( \frac{1}{2}\,\sqrt{-{\eta }}{\phi } \right) }{2{\varrho }}}\), | \(\frac{1}{2}\,{\frac{ \left( 1+ \left( \cot \left( \frac{1}{2}\,\sqrt{-{\eta }}{\phi } \right) \right) ^{2} \right) {\eta }}{{\rho }+\sqrt{-{\eta }}\cot \left( \frac{1}{2}\,\sqrt{-{\eta }}{\phi } \right) }}\), | |||
\(-{\frac{{\rho }}{2{\varrho }}}+{\frac{\sqrt{-{\eta }} \left( \tan \left( \sqrt{-{\eta }}{\phi } \right) + \left( \sec \left( \sqrt{-{\eta }}{\phi } \right) \right) \right) }{2{\varrho }}}\), | \(-{\frac{{\eta }\, \left( 1+\sin \left( \sqrt{-{\eta }}{\phi } \right) \right) \sec \left( \sqrt{-{\eta }}{\phi } \right) }{ -{\rho }\cos \left( \sqrt{-{\eta }}{\phi } \right) +\sqrt{-{\eta }}\sin \left( \sqrt{-{\eta }}{\phi } \right) +\sqrt{-{\eta }} }}\), | |||
\(-{\frac{{\rho }}{2{\varrho }}}+{\frac{\sqrt{-{\eta }} \left( \tan \left( \sqrt{-{\eta }}{\phi } \right) - \left( \sec \left( \sqrt{-{\eta }}{\phi } \right) \right) \right) }{2{\varrho }}}\). | \({\frac{{\eta }\, \left( \sin \left( \sqrt{-{\eta }}{\phi } \right) -1 \right) \sec \left( \sqrt{-{\eta }}{\phi } \right) }{ -{\rho }\cos \left( \sqrt{-{\eta }}{\phi } \right) +\sqrt{-{\eta }}\sin \left( \sqrt{-{\eta }}{\phi } \right) -\sqrt{-{\eta }} }}\). | |||
2 | Hyperbolic Solutions | \({\eta }>0, \quad {\varrho }\ne 0\) | \(-{\frac{{\rho }}{2{\varrho }}}-{\frac{\sqrt{{\eta }}\tanh \left( \frac{1}{2}\,\sqrt{{\eta }}{\phi } \right) }{2{\varrho }}}\), | \(-\frac{1}{2}\,{\frac{ \left( -1+ \left( \tanh \left( \frac{1}{2}\,\sqrt{{\eta }}{\phi } \right) \right) ^{2} \right) {\eta }}{{\rho }+\sqrt{{\eta }}\tanh \left( \frac{1}{2}\,\sqrt{{\eta }}{\phi } \right) }}\), |
\(-{\frac{{\rho }}{2{\varrho }}}-{\frac{\sqrt{{\eta }} \left( \tanh \left( \sqrt{{\eta }}{\phi } \right) + i \left( \textrm{sech} \left( \sqrt{{\eta }}{\phi } \right) \right) \right) }{2{\varrho }}}\), | \(-{\frac{{\eta }\, \left( -1+i\sinh \left( \sqrt{{\eta }}{\phi } \right) \right) }{\cosh \left( \sqrt{{\eta }}{\phi } \right) \left( {\rho }\cosh \left( \sqrt{{\eta }}{\phi } \right) +\sqrt{{\eta }}\sinh \left( \sqrt{ {\eta }}{\phi } \right) +i\sqrt{{\eta }} \right) }}\), | |||
\(-{\frac{{\rho }}{2{\varrho }}}-{\frac{\sqrt{{\eta }} \left( \tanh \left( \sqrt{{\eta }}{\phi } \right) - i \left( \textrm{sech} \left( \sqrt{{\eta }}{\phi } \right) \right) \right) }{2{\varrho }}}\), | \(-{\frac{{\eta }\, \left( 1+i\sinh \left( \sqrt{{\eta }}{\phi } \right) \right) }{\cosh \left( \sqrt{{\eta }}{\phi } \right) \left( -{\rho }\cosh \left( \sqrt{{\eta }}{\phi } \right) -\sqrt{{\eta }}\sinh \left( \sqrt{ {\eta }}{\phi } \right) +i\sqrt{{\eta }} \right) }},\) | |||
\(-{\frac{{\rho }}{2{\varrho }}}-{\frac{\sqrt{{\eta }} \left( \coth \left( \sqrt{{\eta }}{\phi } \right) + \left( \textrm{csch} \left( \sqrt{{\eta }}{\phi } \right) \right) \right) }{2{\varrho }}}\). | \(-\frac{1}{4}\,{\frac{{\eta }\, \left( 2\, \left( \cosh \left( \frac{1}{4}\,\sqrt{ {\eta }}{\phi } \right) \right) ^{2}-1 \right) }{{\Psi } \left( -2\,{\rho }{\Psi } +\sqrt{{\eta }} \right) }}.\) | |||
3 | Rational Solutions | \({\eta }=0\) | \(-2\,{\frac{{\varpi } \left( {\rho }{\phi }\,+2 \right) }{{{\rho }}^{2}{\phi }\,}}\), | \(-2\,{\frac{1}{{\phi }\, \left( {\rho }{\phi }+2 \right) }}\), |
\({\eta }=0\), & \({\rho }={\varrho }=0\) | \({\phi }\,{\varpi }\), | \(\frac{1}{{\phi }\,}\), | ||
\({\eta }=0\), & \({\rho }={\varpi }=0\) | \(-{\frac{1}{{\phi }\,{\varrho }}}\). | \(-\frac{1}{{\phi }\,}\). | ||
4 | Exponential Solutions | \({\varrho }=0\), & \({\rho }={{\sigma }}\), \({\varpi }={\varsigma }{{\sigma }}\) | \({e}^{{{\sigma }}\,{\phi }}-{\varsigma }\), | \({\frac{{{\sigma }}\,{\textrm{e}^{{{\sigma }}\,{\phi }}}}{{\textrm{e}^{{{\sigma }}\,{\phi }}}-{\varsigma }}}\), |
\({\varpi }=0\), & \({\rho }={{\sigma }}\), \({\varrho }={\varsigma }{{\sigma }}\) | \({\frac{{e}^{{{\sigma }}\,{\phi }}}{1-{\varsigma }{e}^{{{\sigma }} {\phi }}}}\). | \(-{\frac{{{\sigma }}}{-1+{\varsigma }{\textrm{e}^{{{\sigma }}\,{\phi }}}}}\). | ||
5 | Rational-Hyperbolic Solutions | \({\varpi }=0\), & \({\rho }\ne 0\), \({\varrho }\ne 0\) | \(-{\frac{{s_1}\,{\rho }}{{\varrho } \left( \cosh \left( {\rho } {\phi } \right) -\sinh \left( {\rho } {\phi } \right) +{s_1} \right) }}\), | \({\frac{{\rho } \left( \sinh \left( {\rho }{\phi } \right) -\cosh \left( {\rho }{\phi } \right) \right) }{-\cosh \left( {\rho }{\phi } \right) +\sinh \left( {\rho }{\phi } \right) -{s_1}}}\), |
\(-{\frac{{\rho } \left( \cosh \left( {\rho } {\phi } \right) +\sinh \left( {\rho } {\phi } \right) \right) }{{\varrho } \left( \cosh \left( {\rho } {\phi } \right) +\sinh \left( {\rho } {\phi } \right) +{s_2} \right) }}\). | \({\frac{{\rho }{s_2}}{\cosh \left( {\rho }{\phi } \right) +\sinh \left( {\rho }{\phi } \right) + {s_2}}}\). |
