Internal problem ID [10459]
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]
\[ \boxed {y^{\prime }-y^{2}=-a^{2}+a \lambda \sinh \left (\lambda x \right )-a^{2} \sinh \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 508
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 \left (x \right ) = \frac {\cosh \left (\lambda x \right ) \left (i \operatorname {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 \operatorname {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 (\operatorname {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}\, c_{1} +\operatorname {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 a c_{1} i-c_{1} \lambda \right ) \operatorname {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 \operatorname {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}-\operatorname {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 a \operatorname {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}\right ) \cosh \left (\lambda x \right )}{2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (\operatorname {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}\, c_{1} +\operatorname {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: 11.807 (sec). Leaf size: 162
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 \frac {e^{\lambda (-x)} \left (a \left (e^{2 \lambda x}+1\right ) \int _1^{e^{x \lambda }}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]-2 \lambda e^{\frac {a e^{\lambda (-x)} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}+a c_1 e^{2 \lambda x}+a c_1\right )}{2 \left (\int _1^{e^{x \lambda }}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]+c_1\right )} y(x)\to \frac {1}{2} a e^{\lambda (-x)} \left (e^{2 \lambda x}+1\right ) \end{align*}