Internal problem ID [9775]
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]
Solve \begin {gather*} \boxed {2 y^{\prime }-\left (\lambda +a -\cos \left (\lambda x \right ) a \right ) y^{2}-\lambda +a +\cos \left (\lambda x \right ) a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \relax (x ) = \left (-\frac {2 c_{1} \lambda \,{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }}}{\left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}} \left (\left (\int -\frac {{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (-\lambda -a +a \cos \left (\lambda x \right )\right ) \lambda \sin \left (\lambda x \right )}{\sqrt {\cos \left (\lambda x \right )+1}\, \left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}}}d x \right ) c_{1}+1\right ) \sqrt {\cos \left (\lambda x \right )+1}}+\frac {\left (\sqrt {\cos \left (\lambda x \right )+1}\, \left (\int -\frac {{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (-\lambda -a +a \cos \left (\lambda x \right )\right ) \lambda \sin \left (\lambda x \right )}{\sqrt {\cos \left (\lambda x \right )+1}\, \left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}}}d x \right ) c_{1}+\sqrt {\cos \left (\lambda x \right )+1}\right ) \cos \left (\lambda x \right )-\sqrt {\cos \left (\lambda x \right )+1}\, \left (\int -\frac {{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (-\lambda -a +a \cos \left (\lambda x \right )\right ) \lambda \sin \left (\lambda x \right )}{\sqrt {\cos \left (\lambda x \right )+1}\, \left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}}}d x \right ) c_{1}-\sqrt {\cos \left (\lambda x \right )+1}}{\left (\left (\int -\frac {{\mathrm e}^{\frac {a \cos \left (\lambda x \right )}{\lambda }} \left (-\lambda -a +a \cos \left (\lambda x \right )\right ) \lambda \sin \left (\lambda x \right )}{\sqrt {\cos \left (\lambda x \right )+1}\, \left (\cos \left (\lambda x \right )-1\right )^{\frac {3}{2}}}d x \right ) c_{1}+1\right ) \sqrt {\cos \left (\lambda x \right )+1}\, \left (\cos \left (\lambda x \right )-1\right )^{2}}\right ) \sin \left (\lambda x \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[2*y'[x]==(\[Lambda]+a-a*Cos[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Cos[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
Not solved