Internal problem ID [10455]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.4-1. Equations with hyperbolic
sine and cosine
Problem number: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (a \sinh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} \sinh \left (\lambda x \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 250
dsolve((a*sinh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*sinh(lambda*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {4 \left (\left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) a \left (\cosh \left (\frac {x \lambda }{2}\right )^{2}-\frac {1}{2}\right ) \sqrt {a^{2}+b^{2}}+\frac {\left (a^{2}+b^{2}\right )^{2} \left (a^{2} \cosh \left (\frac {x \lambda }{2}\right )^{2}+a b \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right )}{2}\right ) \lambda }{\sqrt {a^{2}+b^{2}}\, \left (2 a \cosh \left (\frac {x \lambda }{2}\right ) \left (a^{2}+b^{2}\right )^{\frac {3}{2}} \left (a \sinh \left (\frac {x \lambda }{2}\right )+b \cosh \left (\frac {x \lambda }{2}\right )\right )+4 \left (\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2}+\operatorname {arctanh}\left (\frac {-\tanh \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {a^{2}+b^{2}}}\right ) b^{4}-c_{1} \right ) \left (\sinh \left (\frac {x \lambda }{2}\right ) a \cosh \left (\frac {x \lambda }{2}\right )+\frac {b}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 24.532 (sec). Leaf size: 202
DSolve[(a*Sinh[\[Lambda]*x]+b)*(y'[x]-y[x]^2)+a*\[Lambda]^2*Sinh[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\lambda \left (\sqrt {-a^2-b^2} (b-a \sinh (\lambda x))+a \cosh (\lambda x) \left (2 b \arctan \left (\frac {a-b \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-c_1 \lambda \left (-a^2-b^2\right )^{3/2}\right )\right )}{-a \sqrt {-a^2-b^2} \cosh (\lambda x)+(a \sinh (\lambda x)+b) \left (2 b \arctan \left (\frac {a-b \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-c_1 \lambda \left (-a^2-b^2\right )^{3/2}\right )} \\ y(x)\to -\frac {a \lambda \cosh (\lambda x)}{a \sinh (\lambda x)+b} \\ \end{align*}