Internal problem ID [9727]
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: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda a +a \left (a +\lambda \right ) \left (\coth ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 203
dsolve(diff(y(x),x)=y(x)^2+a*lambda-a*(a+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\left (-c_{1} a -c_{1} \lambda \right ) \LegendreQ \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\left (-a -\lambda \right ) \LegendreP \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )\right ) \coth \left (\lambda x \right )}{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )}+\frac {c_{1} \lambda \LegendreQ \left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )}{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )}+\frac {\LegendreP \left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right ) \lambda }{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 4.061 (sec). Leaf size: 156
DSolve[y'[x]==y[x]^2+a*\[Lambda]-a*(a+\[Lambda])*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a e^{\lambda x} (\coth (\lambda x)-1) \left (2 \lambda \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }} \sinh (\lambda x)+\cosh (\lambda x) \left (\lambda \, _2F_1\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda };1-\frac {a}{\lambda };e^{2 x \lambda }\right )-2 a c_1 \left (e^{\lambda x}\right )^{\frac {2 a}{\lambda }}\right )\right )}{-\lambda \, _2F_1\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda };1-\frac {a}{\lambda };e^{2 x \lambda }\right )+2 a c_1 \left (e^{\lambda x}\right )^{\frac {2 a}{\lambda }}} \\ y(x)\to a \coth (\lambda x) \\ \end{align*}