Internal problem ID [9732]
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: 27.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \lambda -b \lambda +2 a b +a \left (a +\lambda \right ) \left (\tanh ^{2}\left (\lambda x \right )\right )+b \left (b +\lambda \right ) \left (\coth ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 1352
dsolve(diff(y(x),x)=y(x)^2+a*lambda+b*lambda-2*a*b-a*(a+lambda)*tanh(lambda*x)^2-b*(b+lambda)*coth(lambda*x)^2,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 17.945 (sec). Leaf size: 162
DSolve[y'[x]==y[x]^2+a*\[Lambda]+b*\[Lambda]-2*a*b-a*(a+\[Lambda])*Tanh[\[Lambda]*x]^2-b*(b+\[Lambda])*Coth[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (\frac {4 \lambda (a+b) \left (e^{2 \lambda x}+1\right )^{\frac {2 a}{\lambda }} \left (1-e^{2 \lambda x}\right )^{\frac {2 b}{\lambda }}}{\lambda F_1\left (-\frac {a+b}{\lambda };-\frac {2 b}{\lambda },-\frac {2 a}{\lambda };-\frac {a+b-\lambda }{\lambda };e^{2 x \lambda },-e^{2 x \lambda }\right )-c_1 (a+b) \left (e^{2 \lambda x}\right )^{\frac {a+b}{\lambda }}}+2 a \tanh (\lambda x)+b \tanh \left (\frac {\lambda x}{2}\right )+b \coth \left (\frac {\lambda x}{2}\right )\right ) \\ y(x)\to a \tanh (\lambda x)+b \coth (\lambda x) \\ \end{align*}