Internal problem ID [10476]
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: 18.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=\lambda a -a \left (a +\lambda \right ) \tanh \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 198
dsolve(diff(y(x),x)=y(x)^2+a*lambda-a*(a+lambda)*tanh(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (c_{1} a +c_{1} \lambda \right ) \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\left (a +\lambda \right ) \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \tanh \left (\lambda x \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )}+\frac {c_{1} \lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )}+\frac {\operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda }{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a}{\lambda }, \tanh \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 8.574 (sec). Leaf size: 177
DSolve[y'[x]==y[x]^2+a*\[Lambda]-a*(a+\[Lambda])*Tanh[\[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 (e^{2 \lambda x}+1\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*}