Internal problem ID [9718]
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]
\[ \boxed {\left (a \cosh \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} \cosh \left (\lambda x \right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 758
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 (\left (\left (\left (2 \sqrt {a^{2}-b^{2}}\, a^{3} b -4 \sqrt {a^{2}-b^{2}}\, a^{2} b^{2}+2 \sqrt {a^{2}-b^{2}}\, a \,b^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )-2 \sqrt {a^{2}-b^{2}}\, c_{1} a^{3}+4 \sqrt {a^{2}-b^{2}}\, c_{1} a^{2} b -2 \sqrt {a^{2}-b^{2}}\, c_{1} a \,b^{2}\right ) \sinh \left (\lambda x \right )+\left (-a^{5}+2 a^{3} b^{2}-a \,b^{4}\right ) \cosh \left (\lambda x \right )-a^{4} b +2 b^{3} a^{2}-b^{5}\right ) \tanh \left (\frac {\lambda x}{2}\right )^{4}+\left (-2 a^{5}+2 a^{4} b +2 a^{3} b^{2}-2 b^{3} a^{2}\right ) \sinh \left (\lambda x \right ) \tanh \left (\frac {\lambda x}{2}\right )^{3}+\left (\left (\left (4 \sqrt {a^{2}-b^{2}}\, a^{3} b -4 \sqrt {a^{2}-b^{2}}\, a \,b^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )-4 \sqrt {a^{2}-b^{2}}\, c_{1} a^{3}+4 \sqrt {a^{2}-b^{2}}\, c_{1} a \,b^{2}\right ) \sinh \left (\lambda x \right )+\left (2 a^{5}-4 a^{3} b^{2}+2 a \,b^{4}\right ) \cosh \left (\lambda x \right )+2 a^{4} b -4 b^{3} a^{2}+2 b^{5}\right ) \tanh \left (\frac {\lambda x}{2}\right )^{2}+\left (-2 a^{5}-2 a^{4} b +2 a^{3} b^{2}+2 b^{3} a^{2}\right ) \sinh \left (\lambda x \right ) \tanh \left (\frac {\lambda x}{2}\right )+\left (\left (2 \sqrt {a^{2}-b^{2}}\, a^{3} b +4 \sqrt {a^{2}-b^{2}}\, a^{2} b^{2}+2 \sqrt {a^{2}-b^{2}}\, a \,b^{3}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )-2 \sqrt {a^{2}-b^{2}}\, c_{1} a^{3}-4 \sqrt {a^{2}-b^{2}}\, c_{1} a^{2} b -2 \sqrt {a^{2}-b^{2}}\, c_{1} a \,b^{2}\right ) \sinh \left (\lambda x \right )+\left (-a^{5}+2 a^{3} b^{2}-a \,b^{4}\right ) \cosh \left (\lambda x \right )-a^{4} b +2 b^{3} a^{2}-b^{5}\right )}{2 \left (\left (\left (-b a +b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )+c_{1} a -c_{1} b \right ) \tanh \left (\frac {\lambda x}{2}\right )^{2}+a \sqrt {a^{2}-b^{2}}\, \tanh \left (\frac {\lambda x}{2}\right )+\left (-b a -b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )+c_{1} a +c_{1} b \right ) \left (\cosh \left (\lambda x \right ) a +b \right ) \sqrt {a^{2}-b^{2}}\, \left (a +b +\left (a -b \right ) \tanh \left (\frac {\lambda x}{2}\right )^{2}\right )} \]
✓ Solution by Mathematica
Time used: 5.573 (sec). Leaf size: 242
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 (2 a b \sinh (\lambda x) \arctan \left (\frac {(b-a) \tanh \left (\frac {\lambda x}{2}\right )}{\sqrt {(a-b) (a+b)}}\right )+a \sqrt {(a-b) (a+b)} (\cosh (\lambda x)+c_1 \lambda (a-b) (a+b) \sinh (\lambda x))-b \sqrt {(a-b) (a+b)}\right )}{-a \sqrt {a^2-b^2} \sinh (\lambda x)+2 b^2 \cot ^{-1}\left (\frac {(a+b) \coth \left (\frac {\lambda x}{2}\right )}{\sqrt {(a-b) (a+b)}}\right )-b c_1 \lambda ((a-b) (a+b))^{3/2}+a \cosh (\lambda x) \left (2 b \cot ^{-1}\left (\frac {(a+b) \coth \left (\frac {\lambda x}{2}\right )}{\sqrt {(a-b) (a+b)}}\right )-c_1 \lambda ((a-b) (a+b))^{3/2}\right )} \\ y(x)\to -\frac {a \lambda \sinh (\lambda x)}{a \cosh (\lambda x)+b} \\ \end{align*}