Internal problem ID [10481]
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: 23.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=3 \lambda a -\lambda ^{2}-a \left (a +\lambda \right ) \coth \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 240
dsolve(diff(y(x),x)=y(x)^2-lambda^2+3*a*lambda-a*(a+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (c_{1} a +c_{1} \lambda \right ) \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\left (a +\lambda \right ) \operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )\right ) \coth \left (\lambda x \right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}+\frac {2 c_{1} \lambda \operatorname {LegendreQ}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}+\frac {2 \operatorname {LegendreP}\left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right ) \lambda }{c_{1} \operatorname {LegendreQ}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\operatorname {LegendreP}\left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 14.312 (sec). Leaf size: 659
DSolve[y'[x]==y[x]^2-\[Lambda]^2+3*a*\[Lambda]-a*(a+\[Lambda])*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\lambda (a-2 \lambda ) \left (e^{2 \lambda x}+1\right ) \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }} \left (\frac {e^{2 \lambda x}}{e^{2 \lambda x}+1}\right )^{a/\lambda } \left (a \left (-4 e^{2 \lambda x}+e^{4 \lambda x}-1\right )+\lambda -\lambda e^{4 \lambda x}\right ) \operatorname {AppellF1}\left (1-\frac {a}{\lambda },-\frac {2 a}{\lambda },\frac {a}{\lambda },2-\frac {a}{\lambda },\frac {2}{1+e^{2 x \lambda }},\frac {1}{1+e^{2 x \lambda }}\right )+(a-\lambda ) \left (8 a \lambda e^{2 \lambda x} \left (\frac {e^{2 \lambda x}}{e^{2 \lambda x}+1}\right )^{a/\lambda } \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }+1} \operatorname {AppellF1}\left (2-\frac {a}{\lambda },1-\frac {2 a}{\lambda },\frac {a}{\lambda },3-\frac {a}{\lambda },\frac {2}{1+e^{2 x \lambda }},\frac {1}{1+e^{2 x \lambda }}\right )-2 a \lambda e^{2 \lambda x} \left (\frac {e^{2 \lambda x}}{e^{2 \lambda x}+1}\right )^{a/\lambda } \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }+1} \operatorname {AppellF1}\left (2-\frac {a}{\lambda },-\frac {2 a}{\lambda },\frac {a+\lambda }{\lambda },3-\frac {a}{\lambda },\frac {2}{1+e^{2 x \lambda }},\frac {1}{1+e^{2 x \lambda }}\right )+c_1 (a-2 \lambda ) \left (e^{2 \lambda x}+1\right )^2 \left (e^{2 \lambda x}\right )^{a/\lambda } \left (\frac {e^{2 \lambda x}-1}{e^{2 \lambda x}+1}\right )^{\frac {2 a}{\lambda }} \left (a \left (e^{2 \lambda x}+1\right )^2-\lambda \left (e^{2 \lambda x}-1\right )^2\right )\right )}{(2 \lambda -a) \left (e^{2 \lambda x}-1\right ) \left (e^{2 \lambda x}+1\right )^2 \left (-\lambda \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }} \left (\frac {e^{2 \lambda x}}{e^{2 \lambda x}+1}\right )^{a/\lambda } \operatorname {AppellF1}\left (1-\frac {a}{\lambda },-\frac {2 a}{\lambda },\frac {a}{\lambda },2-\frac {a}{\lambda },\frac {2}{1+e^{2 x \lambda }},\frac {1}{1+e^{2 x \lambda }}\right )+c_1 (\lambda -a) \left (e^{2 \lambda x}+1\right ) \left (e^{2 \lambda x}\right )^{a/\lambda } \left (\frac {e^{2 \lambda x}-1}{e^{2 \lambda x}+1}\right )^{\frac {2 a}{\lambda }}\right )} y(x)\to \frac {a \left (e^{2 \lambda x}+1\right )^2-\lambda \left (e^{2 \lambda x}-1\right )^2}{e^{4 \lambda x}-1} \end{align*}