Internal problem ID [9753]
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]
\[ \boxed {y^{\prime }-y^{2}+a^{2}-a \lambda \sin \left (\lambda x \right )-a^{2} \sin \left (\lambda x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 415
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)
\[ y \left (x \right ) = -\frac {\left (\left (4 \sqrt {-\cos \left (\lambda x \right )^{2}+1}\, c_{1} a +4 c_{1} a +2 c_{1} \lambda \right ) \operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sqrt {-\cos \left (\lambda x \right )^{2}+1}}{2}+\frac {1}{2}\right )+2 a \operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sqrt {-\cos \left (\lambda x \right )^{2}+1}}{2}+\frac {1}{2}\right ) \sqrt {2 \sqrt {-\cos \left (\lambda x \right )^{2}+1}+2}+\left (2 \sqrt {-\cos \left (\lambda x \right )^{2}+1}\, c_{1} \lambda +2 c_{1} \lambda \right ) \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sqrt {-\cos \left (\lambda x \right )^{2}+1}}{2}+\frac {1}{2}\right )+\lambda \operatorname {HeunCPrime}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sqrt {-\cos \left (\lambda x \right )^{2}+1}}{2}+\frac {1}{2}\right ) \sqrt {2 \sqrt {-\cos \left (\lambda x \right )^{2}+1}+2}\right ) \cos \left (\lambda x \right ) \sin \left (\lambda x \right )}{2 \sqrt {2 \sqrt {-\cos \left (\lambda x \right )^{2}+1}+2}\, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\, \left (\operatorname {HeunC}\left (\frac {4 a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sqrt {-\cos \left (\lambda x \right )^{2}+1}}{2}+\frac {1}{2}\right ) \sqrt {2 \sqrt {-\cos \left (\lambda x \right )^{2}+1}+2}\, c_{1} +\operatorname {HeunC}\left (\frac {4 a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, -\frac {2 a}{\lambda }, \frac {8 a +3 \lambda }{8 \lambda }, \frac {\sqrt {-\cos \left (\lambda x \right )^{2}+1}}{2}+\frac {1}{2}\right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2-a^2+a*\[Lambda]*Sin[\[Lambda]*x]+a^2*Sin[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved