Internal problem ID [9727]
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]
\[ \boxed {y^{\prime }-y^{2}+2 \lambda ^{2} \tanh \left (\lambda x \right )^{2}+2 \lambda ^{2} \coth \left (\lambda x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 330
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 \left (x \right ) = \frac {\lambda \left (\left (\left (-3 \coth \left (\lambda x \right )^{2} c_{1} -c_{1} \right ) \operatorname {csch}\left (\lambda x \right )^{2} \sinh \left (\lambda x \right )^{2}+2 \coth \left (\lambda x \right ) \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2} c_{1} \right ) \ln \left (\coth \left (\lambda x \right )+1\right )+\left (\left (3 \coth \left (\lambda x \right )^{2} c_{1} +c_{1} \right ) \operatorname {csch}\left (\lambda x \right )^{2} \sinh \left (\lambda x \right )^{2}-2 \coth \left (\lambda x \right ) \cosh \left (\lambda x \right ) \sinh \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2} c_{1} \right ) \ln \left (\coth \left (\lambda x \right )-1\right )+\left (-12 \cosh \left (\lambda x \right )^{2} c_{1} +2 c_{1} \right ) \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2} \sinh \left (\lambda x \right )^{4}+\left (\left (12 \cosh \left (\lambda x \right )^{3} c_{1} -6 \cosh \left (\lambda x \right ) c_{1} \right ) \coth \left (\lambda x \right )^{2}+4 \cosh \left (\lambda x \right )^{3} c_{1} -2 \cosh \left (\lambda x \right ) c_{1} \right ) \operatorname {csch}\left (\lambda x \right )^{2} \sinh \left (\lambda x \right )^{3}+\left (-12 \cosh \left (\lambda x \right )^{4} c_{1} +6 \cosh \left (\lambda x \right )^{2} c_{1} +2 c_{1} \right ) \coth \left (\lambda x \right ) \operatorname {csch}\left (\lambda x \right )^{2} \sinh \left (\lambda x \right )^{2}+\coth \left (\lambda x \right )^{2}+1\right )}{\coth \left (\lambda x \right ) \left (-\sinh \left (\lambda x \right )^{2} \operatorname {csch}\left (\lambda x \right )^{2} \ln \left (\coth \left (\lambda x \right )+1\right ) c_{1} +\sinh \left (\lambda x \right )^{2} \operatorname {csch}\left (\lambda x \right )^{2} \ln \left (\coth \left (\lambda x \right )-1\right ) c_{1} +1+\left (4 \cosh \left (\lambda x \right )^{3} c_{1} -2 \cosh \left (\lambda x \right ) c_{1} \right ) \operatorname {csch}\left (\lambda x \right )^{2} \sinh \left (\lambda x \right )^{3}\right )} \]
✓ Solution by Mathematica
Time used: 3.746 (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*}