Internal problem ID [10528]
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]
\[ \boxed {2 y^{\prime }-\left (\lambda +a -a \cos \left (\lambda x \right )\right ) y^{2}=-a +\lambda -a \cos \left (\lambda x \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 307
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)
\[ y \left (x \right ) = \left (-\frac {2 c_{1} \lambda \,{\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }}}{\left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}} \left (\left (\int -\frac {\left (-\lambda -a +\cos \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \lambda \sin \left (\lambda x \right )}{\left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\cos \left (\lambda x \right )+1}}d x \right ) c_{1} +1\right ) \sqrt {\cos \left (\lambda x \right )+1}}+\frac {\left (\left (\int -\frac {\left (-\lambda -a +\cos \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \lambda \sin \left (\lambda x \right )}{\left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\cos \left (\lambda x \right )+1}}d x \right ) \sqrt {\cos \left (\lambda x \right )+1}\, c_{1} +\sqrt {\cos \left (\lambda x \right )+1}\right ) \cos \left (\lambda x \right )-\left (\int -\frac {\left (-\lambda -a +\cos \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \lambda \sin \left (\lambda x \right )}{\left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\cos \left (\lambda x \right )+1}}d x \right ) \sqrt {\cos \left (\lambda x \right )+1}\, c_{1} -\sqrt {\cos \left (\lambda x \right )+1}}{\left (\left (\int -\frac {\left (-\lambda -a +\cos \left (\lambda x \right ) a \right ) {\mathrm e}^{\frac {\cos \left (\lambda x \right ) a}{\lambda }} \lambda \sin \left (\lambda x \right )}{\left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}} \sqrt {\cos \left (\lambda x \right )+1}}d x \right ) c_{1} +1\right ) \sqrt {\cos \left (\lambda x \right )+1}\, \left (\cos \left (\lambda x \right )-1\right )^{2}}\right ) \sin \left (\lambda x \right ) \]
✓ 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*}