Internal problem ID [10500]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with
sine
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=-a^{2}+a \lambda \sin \left (\lambda x \right )+a^{2} \sin \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 289
dsolve(diff(y(x),x)=y(x)^2-a^2+a*lambda*sin(lambda*x)+a^2*sin(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-2 a c_{1} \cos \left (x \lambda \right ) \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )-c_{1} \lambda \cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )-2 \left (a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )+\frac {\lambda \left (\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right ) c_{1} \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )+\operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )\right )}{2}\right ) \cos \left (x \lambda \right )}{2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) c_{1} +2 \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sin \left (x \lambda \right )}{2}+\frac {1}{2}\right )} \]
✓ Solution by Mathematica
Time used: 4.337 (sec). Leaf size: 132
DSolve[y'[x]==y[x]^2-a^2+a*\[Lambda]*Sin[\[Lambda]*x]+a^2*Sin[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a c_1 \cos (\lambda x) \int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]+a \cos (\lambda x)+c_1 e^{-\frac {2 a \sin (\lambda x)}{\lambda }}}{1+c_1 \int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]} \\ y(x)\to -\frac {e^{-\frac {2 a \sin (\lambda x)}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin (\lambda K[1])}{\lambda }}dK[1]}-a \cos (\lambda x) \\ \end{align*}