Internal problem ID [9706]
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: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+a^{2}-a \lambda \sinh \left (\lambda x \right )+a^{2} \left (\sinh ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 506
dsolve(diff(y(x),x)=y(x)^2-a^2+a*lambda*sinh(lambda*x)-a^2*sinh(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\cosh \left (\lambda x \right ) \left (-i \HeunCPrime \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) c_{1} \lambda +2 \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) c_{1} a \right ) \sinh \left (\lambda x \right )}{2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (\sqrt {\sinh \left (\lambda x \right )+i}\, \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) c_{1}+\HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right )\right )}-\frac {\left (\left (2 i c_{1} a +c_{1} \lambda \right ) \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right )-i \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) \sqrt {\sinh \left (\lambda x \right )+i}+2 a \HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) \sqrt {\sinh \left (\lambda x \right )+i}+\HeunCPrime \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) c_{1} \lambda \right ) \cosh \left (\lambda x \right )}{2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (\sqrt {\sinh \left (\lambda x \right )+i}\, \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right ) c_{1}+\HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, \frac {1}{2}-\frac {i \sinh \left (\lambda x \right )}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 4.937 (sec). Leaf size: 75
DSolve[y'[x]==y[x]^2-a^2+a*\[Lambda]*Sinh[\[Lambda]*x]-a^2*Sinh[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to a \cosh (\lambda x)-\frac {\lambda e^{\frac {2 a \sinh (\lambda x)}{\lambda }}}{\int _1^{e^{x \lambda }}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]+c_1} \\ y(x)\to a \cosh (\lambda x) \\ \end{align*}