Internal problem ID [9731]
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: 26.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+2 \lambda ^{2} \left (\tanh ^{2}\left (\lambda x \right )\right )+2 \lambda ^{2} \left (\coth ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 279
dsolve(diff(y(x),x)=y(x)^2-2*lambda^2*tanh(lambda*x)^2-2*lambda^2*coth(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\lambda \left (\left (\left (4 c_{1} \left (\coth ^{2}\left (\lambda x \right )\right )-4 c_{1}\right ) \sinh \left (\lambda x \right )-8 \coth \left (\lambda x \right ) \cosh \left (\lambda x \right ) c_{1}\right ) \ln \left (\frac {\sinh \left (\lambda x \right )+\cosh \left (\lambda x \right )}{\sinh \left (\lambda x \right )}\right )+\left (\left (-4 c_{1} \left (\coth ^{2}\left (\lambda x \right )\right )+4 c_{1}\right ) \sinh \left (\lambda x \right )+8 \coth \left (\lambda x \right ) \cosh \left (\lambda x \right ) c_{1}\right ) \ln \left (\frac {-\sinh \left (\lambda x \right )+\cosh \left (\lambda x \right )}{\sinh \left (\lambda x \right )}\right )-2 c_{1} \coth \left (\lambda x \right ) \sinh \left (5 \lambda x \right )+6 c_{1} \coth \left (\lambda x \right ) \sinh \left (3 \lambda x \right )+\left (-4 \left (\coth ^{2}\left (\lambda x \right )\right )+8 c_{1} \coth \left (\lambda x \right )+4\right ) \sinh \left (\lambda x \right )+\left (-c_{1} \cosh \left (5 \lambda x \right )+c_{1} \cosh \left (3 \lambda x \right )\right ) \left (\coth ^{2}\left (\lambda x \right )\right )+8 \cosh \left (\lambda x \right ) \coth \left (\lambda x \right )+c_{1} \cosh \left (5 \lambda x \right )-c_{1} \cosh \left (3 \lambda x \right )\right )}{\coth \left (\lambda x \right ) \left (4 \sinh \left (\lambda x \right ) \ln \left (\frac {-\sinh \left (\lambda x \right )+\cosh \left (\lambda x \right )}{\sinh \left (\lambda x \right )}\right ) c_{1}-4 \sinh \left (\lambda x \right ) \ln \left (\frac {\sinh \left (\lambda x \right )+\cosh \left (\lambda x \right )}{\sinh \left (\lambda x \right )}\right ) c_{1}+c_{1} \cosh \left (5 \lambda x \right )-c_{1} \cosh \left (3 \lambda x \right )+4 \sinh \left (\lambda x \right )\right )} \]
✓ Solution by Mathematica
Time used: 3.761 (sec). Leaf size: 65
DSolve[y'[x]==y[x]^2-2*\[Lambda]^2*Tanh[\[Lambda]*x]^2-2*\[Lambda]^2*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \lambda \left (\cosh (4 \lambda x)-\coth (2 \lambda x) \left (-2 \log \left (e^{2 \lambda x}\right )+c_1\right )-3\right )}{-2 \log \left (e^{2 \lambda x}\right )+\sinh (4 \lambda x)+c_1} \\ y(x)\to 2 \lambda \coth (2 \lambda x) \\ \end{align*}