Internal problem ID [10475]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.4-2. Equations with hyperbolic
tangent and cotangent.
Problem number: 27.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=-2 a b +\lambda a +b \lambda -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}-b \left (b +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 289
dsolve(diff(y(x),x)=y(x)^2+a*lambda+b*lambda-2*a*b-a*(a+lambda)*tanh(lambda*x)^2-b*(b+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-4 c_{1} \lambda \left (b -\frac {\lambda }{2}\right ) \coth \left (x \lambda \right )^{\frac {2 a +2 \lambda }{\lambda }} \operatorname {csch}\left (x \lambda \right )^{2} \operatorname {hypergeom}\left (\left [2, -\frac {2 b -3 \lambda }{2 \lambda }\right ], \left [\frac {2 a +5 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right )-2 c_{1} \left (\left (\left (\frac {3 a}{2}+\frac {3 b}{2}\right ) \lambda +a b \right ) \coth \left (x \lambda \right )^{\frac {2 a +2 \lambda }{\lambda }}+\left (-\frac {5 \,\operatorname {sech}\left (x \lambda \right ) \lambda \left (a +\frac {3 \lambda }{5}\right ) \operatorname {csch}\left (x \lambda \right )}{2}+a^{2} \tanh \left (x \lambda \right )\right ) \coth \left (x \lambda \right )^{\frac {2 a +\lambda }{\lambda }}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right )+2 \left (a \tanh \left (x \lambda \right )+\coth \left (x \lambda \right ) b \right ) \left (a +\frac {3 \lambda }{2}\right ) \left (-\operatorname {csch}\left (x \lambda \right )^{2}\right )^{\frac {a +b}{\lambda }}}{\left (\operatorname {hypergeom}\left (\left [1, \frac {-2 b +\lambda }{2 \lambda }\right ], \left [\frac {2 a +3 \lambda }{2 \lambda }\right ], \coth \left (x \lambda \right )^{2}\right ) c_{1} \coth \left (x \lambda \right )^{\frac {2 a +\lambda }{\lambda }}+\left (-\operatorname {csch}\left (x \lambda \right )^{2}\right )^{\frac {a +b}{\lambda }}\right ) \left (2 a +3 \lambda \right )} \]
✓ Solution by Mathematica
Time used: 40.238 (sec). Leaf size: 493
DSolve[y'[x]==y[x]^2+a*\[Lambda]+b*\[Lambda]-2*a*b-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2-b*(b+\[Lambda])*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {(a+b) \left (e^{2 \lambda x}\right )^{\frac {a+b}{\lambda }} \left (\frac {2 \lambda \left (a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2\right ) \left (e^{2 \lambda x}\right )^{-\frac {a+b}{\lambda }} \operatorname {AppellF1}\left (-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-\lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )}{(a+b) \left (e^{2 \lambda x}-1\right ) \left (e^{2 \lambda x}+1\right )}+4 \lambda \left (e^{2 \lambda x}\right )^{-\frac {a+b}{\lambda }} \operatorname {AppellF1}\left (-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-\lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )+\frac {8 \lambda \left (e^{2 \lambda x}\right )^{-\frac {a+b-\lambda }{\lambda }} \left (a \operatorname {AppellF1}\left (1-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },1-\frac {2 a}{\lambda },-\frac {a+b-2 \lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )-b \operatorname {AppellF1}\left (1-\frac {a+b}{\lambda },1-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-2 \lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )\right )}{-a-b+\lambda }-\frac {2 c_1 \left (a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2\right )}{e^{4 \lambda x}-1}\right )}{2 \left (-\lambda \operatorname {AppellF1}\left (-\frac {a+b}{\lambda },-\frac {2 b}{\lambda },-\frac {2 a}{\lambda },-\frac {a+b-\lambda }{\lambda },e^{2 x \lambda },-e^{2 x \lambda }\right )+c_1 (a+b) \left (e^{2 \lambda x}\right )^{\frac {a+b}{\lambda }}\right )} \\ y(x)\to \frac {a \left (e^{2 \lambda x}-1\right )^2+b \left (e^{2 \lambda x}+1\right )^2}{e^{4 \lambda x}-1} \\ \end{align*}