Internal problem ID [9776]
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: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\left (\lambda +\left (\cos ^{2}\left (\lambda x \right )\right ) a \right ) y^{2}-\lambda +a -\left (\cos ^{2}\left (\lambda x \right )\right ) a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 550
dsolve(diff(y(x),x)=(lambda+a*cos(lambda*x)^2)*y(x)^2+lambda-a+a*cos(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = \left (\frac {\left (4 \cos \left (2 \lambda x \right ) \sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} a \lambda +4 \sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} a \lambda +8 \sqrt {\cos \left (2 \lambda x \right )+1}\, c_{1} \lambda ^{2}\right ) {\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }}}{2 \left (\cos \left (2 \lambda x \right )+1\right )^{2} \sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (\lambda +a \left (\cos ^{2}\left (\lambda x \right )\right )\right ) \left (\left (\int -\frac {2 \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\left (\cos \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cos \left (2 \lambda x \right )}}d x \right ) c_{1}+1\right )}+\frac {\left (\sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (\int -\frac {2 \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\left (\cos \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cos \left (2 \lambda x \right )}}d x \right ) c_{1} a +a \sqrt {-1+\cos \left (2 \lambda x \right )}\right ) \left (\cos ^{2}\left (2 \lambda x \right )\right )+\left (\left (2 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a +2 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} \lambda \right ) \left (\int -\frac {2 \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\left (\cos \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cos \left (2 \lambda x \right )}}d x \right )+2 a \sqrt {-1+\cos \left (2 \lambda x \right )}+2 \lambda \sqrt {-1+\cos \left (2 \lambda x \right )}\right ) \cos \left (2 \lambda x \right )+\left (\sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} a +2 \sqrt {-1+\cos \left (2 \lambda x \right )}\, c_{1} \lambda \right ) \left (\int -\frac {2 \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\left (\cos \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cos \left (2 \lambda x \right )}}d x \right )+a \sqrt {-1+\cos \left (2 \lambda x \right )}+2 \lambda \sqrt {-1+\cos \left (2 \lambda x \right )}}{2 \left (\cos \left (2 \lambda x \right )+1\right )^{2} \sqrt {-1+\cos \left (2 \lambda x \right )}\, \left (\lambda +a \left (\cos ^{2}\left (\lambda x \right )\right )\right ) \left (\left (\int -\frac {2 \,{\mathrm e}^{-\frac {a \cos \left (2 \lambda x \right )}{2 \lambda }} \left (a \cos \left (2 \lambda x \right )+a +2 \lambda \right ) \sin \left (2 \lambda x \right ) \lambda }{\left (\cos \left (2 \lambda x \right )+1\right )^{\frac {3}{2}} \sqrt {-1+\cos \left (2 \lambda x \right )}}d x \right ) c_{1}+1\right )}\right ) \sin \left (2 \lambda x \right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==(\[Lambda]+a*Cos[\[Lambda]*x]^2)*y[x]^2+\[Lambda]-a+a*Cos[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved