Internal problem ID [9771]
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]
\[ \boxed {2 y^{\prime }-\left (\lambda +a -\cos \left (\lambda x \right ) a \right ) y^{2}-\lambda +a +\cos \left (\lambda x \right ) a=0} \]
✓ 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: 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