Internal problem ID [9724]
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: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-3 \lambda a +\lambda ^{2}+a \left (a +\lambda \right ) \left (\tanh ^{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+3*a*lambda-lambda^2-a*(a+lambda)*tanh(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\left (-c_{1} a -c_{1} \lambda \right ) \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\left (-a -\lambda \right ) \LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )\right ) \tanh \left (\lambda x \right )}{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}+\frac {2 c_{1} \lambda \LegendreQ \left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )}+\frac {2 \LegendreP \left (\frac {a +\lambda }{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right ) \lambda }{c_{1} \LegendreQ \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )+\LegendreP \left (\frac {a}{\lambda }, \frac {a -\lambda }{\lambda }, \tanh \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 6.974 (sec). Leaf size: 341
DSolve[y'[x]==y[x]^2+3*a*\[Lambda]-\[Lambda]^2-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 e^{2 \lambda x} \left (e^{2 \lambda x}-1\right )^2 ((\lambda -a) \cosh (2 \lambda x)+a+\lambda ) \int _1^{e^{x \lambda }}\frac {K[1]^{1-\frac {2 a}{\lambda }} \left (K[1]^2+1\right )^{\frac {2 a}{\lambda }}}{\left (K[1]^2-1\right )^2}dK[1]+4 e^{4 \lambda x} \sinh (\lambda x) \left (\lambda \left (-\left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }}\right ) \left (e^{\lambda x}\right )^{-\frac {2 a}{\lambda }} \cosh (\lambda x)-2 c_1 \sinh (\lambda x) ((\lambda -a) \cosh (2 \lambda x)+a+\lambda )\right )}{\left (e^{2 \lambda x}-1\right )^3 \left (e^{2 \lambda x}+1\right ) \left (\int _1^{e^{x \lambda }}\frac {K[1]^{1-\frac {2 a}{\lambda }} \left (K[1]^2+1\right )^{\frac {2 a}{\lambda }}}{\left (K[1]^2-1\right )^2}dK[1]+c_1\right )} \\ y(x)\to a \tanh (\lambda x)-\lambda \coth (\lambda x) \\ y(x)\to -\frac {\lambda \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }} \left (e^{\lambda x}\right )^{-\frac {2 a}{\lambda }} \text {csch}^2(\lambda x)}{4 \int _1^{e^{x \lambda }}\frac {K[1]^{1-\frac {2 a}{\lambda }} \left (K[1]^2+1\right )^{\frac {2 a}{\lambda }}}{\left (K[1]^2-1\right )^2}dK[1]}+a \tanh (\lambda x)-\lambda \coth (\lambda x) \\ \end{align*}