Internal problem ID [9728]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+\lambda ^{2}-3 \lambda a +a \left (a +\lambda \right ) \left (\coth ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 245
dsolve(diff(y(x),x)=y(x)^2-lambda^2+3*a*lambda-a*(a+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\left (-c_{1} a -c_{1} \lambda \right ) \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\left (-a -\lambda \right ) \LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )\right ) \coth \left (\lambda x \right )}{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}+\frac {2 c_{1} \lambda \LegendreQ \left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )}+\frac {2 \LegendreP \left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right ) \lambda }{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \coth \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 62.612 (sec). Leaf size: 249
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 {(a-2 \lambda ) e^{3 \lambda x} (\coth (\lambda x)-1) \left (-\lambda e^{\lambda x} F_1\left (1-\frac {a}{\lambda };-\frac {2 a}{\lambda },2;2-\frac {a}{\lambda };e^{2 x \lambda },-e^{2 x \lambda }\right ) ((a-\lambda ) \cosh (2 \lambda x)+a+\lambda )+\frac {2 (a-\lambda ) \left (-\lambda \left (1-e^{2 \lambda x}\right )^{\frac {2 a}{\lambda }} \sinh (\lambda x)+2 c_1 \left (e^{\lambda x}\right )^{\frac {2 a}{\lambda }} \cosh (\lambda x) ((a-\lambda ) \cosh (2 \lambda x)+a+\lambda )\right )}{e^{2 \lambda x}+1}\right )}{(2 \lambda -a) \left (e^{2 \lambda x}+1\right ) \left (\lambda e^{2 \lambda x} F_1\left (1-\frac {a}{\lambda };-\frac {2 a}{\lambda },2;2-\frac {a}{\lambda };e^{2 x \lambda },-e^{2 x \lambda }\right )+2 c_1 (\lambda -a) \left (e^{\lambda x}\right )^{\frac {2 a}{\lambda }}\right )} \\ \end{align*}