Internal problem ID [10449]
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: 318
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)
\[ y \left (x \right ) = \frac {\left (-2 \sinh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right ) \cosh \left (x \lambda \right ) c_{1} a -\cosh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right ) c_{1} \lambda \right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right )+\cosh \left (x \lambda \right ) \left (-2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right ) a +i \lambda \left (\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} \sinh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right )\right )\right )}{2 \sinh \left (\frac {i \pi }{4}+\frac {x \lambda }{2}\right ) \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (x \lambda \right )}{2}+\frac {1}{2}\right )} \]
✓ 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*}