Internal problem ID [10518]
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: 20.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {2 y^{\prime }-\left (\lambda +a -\cos \left (\lambda x \right ) a \right ) y^{2}=-a +\lambda -\cos \left (\lambda x \right ) a} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 122
dsolve(2*diff(y(x),x)=(lambda+a-a*cos(lambda*x))*y(x)^2+lambda-a-a*cos(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\cot \left (\frac {x \lambda }{2}\right ) \lambda \left (\int {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) \left (\csc \left (\frac {x \lambda }{2}\right )^{2} \lambda +2 a \right )d x \right ) c_{1} -2 \csc \left (\frac {x \lambda }{2}\right )^{2} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} c_{1} \lambda +2 i \cot \left (\frac {x \lambda }{2}\right )}{\lambda \left (\int {\mathrm e}^{\frac {a \cos \left (x \lambda \right )}{\lambda }} \operatorname {csgn}\left (\sin \left (\frac {x \lambda }{2}\right )\right ) \left (\csc \left (\frac {x \lambda }{2}\right )^{2} \lambda +2 a \right )d x \right ) c_{1} -2 i} \]
✓ Solution by Mathematica
Time used: 34.139 (sec). Leaf size: 234
DSolve[2*y'[x]==(\[Lambda]+a-a*Cos[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Cos[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \left (c_1 \cot \left (\frac {\lambda x}{2}\right ) \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]+2 c_1 \csc ^2\left (\frac {\lambda x}{2}\right ) e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}+\cot \left (\frac {\lambda x}{2}\right )\right )}{2+2 c_1 \int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]} \\ y(x)\to \frac {1}{2} \csc ^2\left (\frac {\lambda x}{2}\right ) \left (-\frac {4 e^{-\frac {2 a \sin ^2\left (\frac {\lambda x}{2}\right )}{\lambda }}}{\int _1^xe^{-\frac {2 a \sin ^2\left (\frac {1}{2} \lambda K[1]\right )}{\lambda }} \left (\lambda \csc ^2\left (\frac {1}{2} \lambda K[1]\right )+2 a\right )dK[1]}-\sin (\lambda x)\right ) \\ \end{align*}