Internal problem ID [10465]
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: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=-a \,\lambda ^{2} \cosh \left (\lambda x \right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 204
dsolve((a*cosh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*cosh(lambda*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\lambda \left (-2 \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right ) \sqrt {a^{2}-b^{2}}\, a b \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )+2 \sqrt {a^{2}-b^{2}}\, c_{1} a \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )+\left (a +b \right ) \left (\cosh \left (\frac {x \lambda }{2}\right )^{2} a -\frac {a}{2}-\frac {b}{2}\right ) \left (a -b \right )\right )}{\sqrt {a^{2}-b^{2}}\, \left (2 \left (\cosh \left (\frac {x \lambda }{2}\right )^{2} a -\frac {a}{2}+\frac {b}{2}\right ) b \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )-\sqrt {a^{2}-b^{2}}\, a \cosh \left (\frac {x \lambda }{2}\right ) \sinh \left (\frac {x \lambda }{2}\right )-2 c_{1} \left (\cosh \left (\frac {x \lambda }{2}\right )^{2} a -\frac {a}{2}+\frac {b}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 7.749 (sec). Leaf size: 246
DSolve[(a*Cosh[\[Lambda]*x]+b)*(y'[x]-y[x]^2)+a*\[Lambda]^2*Cosh[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\lambda \left (a \sinh (\lambda x) \left (2 b \arctan \left (\frac {(b-a) \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 )+a \sqrt {a^2-b^2} \cosh (\lambda x)+b \left (-\sqrt {a^2-b^2}\right )\right )}{b \left (2 b \arctan \left (\frac {(b-a) \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 )+a \cosh (\lambda x) \left (2 b \arctan \left (\frac {(b-a) \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 )+a \sqrt {a^2-b^2} \sinh (\lambda x)} \\ y(x)\to -\frac {a \lambda \sinh (\lambda x)}{a \cosh (\lambda x)+b} \\ \end{align*}