Internal problem ID [10470]
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]
\[ \boxed {y^{\prime }-y^{2}=\lambda a -a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 122
dsolve(diff(y(x),x)=y(x)^2+a*lambda-a*(a+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right ) \lambda +\operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right ) c_{1} \lambda -\coth \left (x \lambda \right ) \left (c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )\right ) \left (a +\lambda \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \coth \left (x \lambda \right )\right )} \]
✓ Solution by Mathematica
Time used: 8.402 (sec). Leaf size: 175
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 \left (-\lambda \left (e^{2 \lambda x}+1\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },e^{2 x \lambda }\right )+2 \lambda \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }+1}+a c_1 \left (e^{2 \lambda x}+1\right ) \left (e^{2 \lambda x}\right )^{a/\lambda }\right )}{\left (e^{2 \lambda x}-1\right ) \left (-\lambda \operatorname {Hypergeometric2F1}\left (-\frac {2 a}{\lambda },-\frac {a}{\lambda },1-\frac {a}{\lambda },e^{2 x \lambda }\right )+a c_1 \left (e^{2 \lambda x}\right )^{a/\lambda }\right )} \\ y(x)\to \frac {a \left (e^{2 \lambda x}+1\right )}{e^{2 \lambda x}-1} \\ \end{align*}