Internal problem ID [10523]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with
cosine.
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=-a^{2}+\lambda \cos \left (\lambda x \right ) a +a^{2} \cos \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 467
dsolve(diff(y(x),x)=y(x)^2-a^2+a*lambda*cos(lambda*x)+a^2*cos(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \left (\frac {\left (4 \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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{1} a +2 \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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{1} \lambda \right ) \cos \left (\lambda x \right )}{2 \sqrt {2 \cos \left (\lambda x \right )+2}\, \left (\sqrt {2 \cos \left (\lambda x \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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) 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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}+\frac {\left (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 {\cos \left (\lambda x \right )}{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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (\lambda x \right )+2}+\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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \sqrt {2 \cos \left (\lambda x \right )+2}+2 \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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) c_{1} \lambda }{2 \sqrt {2 \cos \left (\lambda x \right )+2}\, \left (\sqrt {2 \cos \left (\lambda x \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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right ) 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 {\cos \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}\right ) \sin \left (\lambda x \right ) \]
✓ Solution by Mathematica
Time used: 3.942 (sec). Leaf size: 131
DSolve[y'[x]==y[x]^2-a^2+a*\[Lambda]*Cos[\[Lambda]*x]+a^2*Cos[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a c_1 \sin (\lambda x) \int _1^xe^{-\frac {2 a \cos (\lambda K[1])}{\lambda }}dK[1]+a \sin (\lambda x)+c_1 \left (-e^{-\frac {2 a \cos (\lambda x)}{\lambda }}\right )}{1+c_1 \int _1^xe^{-\frac {2 a \cos (\lambda K[1])}{\lambda }}dK[1]} y(x)\to a \sin (\lambda x)-\frac {e^{-\frac {2 a \cos (\lambda x)}{\lambda }}}{\int _1^xe^{-\frac {2 a \cos (\lambda K[1])}{\lambda }}dK[1]} \end{align*}