Internal problem ID [9712]
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]
Solve \begin {gather*} \boxed {\left (\sinh \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )+\lambda ^{2} \sinh \left (\lambda x \right ) a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1344
dsolve((a*sinh(lambda*x)+b)*(diff(y(x),x)-y(x)^2)+a*lambda^2*sinh(lambda*x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.337 (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 \text {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 \text {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*}