Internal problem ID [9722]
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]
Solve \begin {gather*} \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} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 931
dsolve((a*cosh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*cosh(lambda*x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.269 (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) \text {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 )}{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 \sqrt {(a-b) (a+b)} \sinh (\lambda x)+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*}