Internal problem ID [9813]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-5. Equations containing
combinations of trigonometric functions.
Problem number: 58.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+\frac {\lambda ^{2}}{2}+\frac {3 \left (\tan ^{2}\left (\lambda x \right )\right ) \lambda ^{2}}{4}-a \left (\cos ^{2}\left (\lambda x \right )\right ) \left (\sin ^{n}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 413
dsolve(diff(y(x),x)=y(x)^2-1/2*lambda^2-3/4*lambda^2*tan(lambda*x)^2+a*cos(lambda*x)^2*sin(lambda*x)^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (-\lambda ^{2} n^{2} c_{1}-4 \lambda ^{2} n c_{1}-3 \lambda ^{2} c_{1}\right ) \hypergeom \left (\left [\right ], \left [\frac {n +3}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right ) \left (\sin ^{3}\left (\lambda x \right )\right )+\left (-\lambda ^{2} n^{2}-4 \lambda ^{2} n -3 \lambda ^{2}\right ) \hypergeom \left (\left [\right ], \left [\frac {n +1}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right ) \left (\sin ^{2}\left (\lambda x \right )\right )+\left (\left (2 \left (\sin ^{2+n}\left (\lambda x \right )\right ) c_{1} a n +2 \left (\sin ^{2+n}\left (\lambda x \right )\right ) c_{1} a \right ) \hypergeom \left (\left [\right ], \left [\frac {2 n +5}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )+\left (-2 \lambda ^{2} n^{2} c_{1}-8 \lambda ^{2} n c_{1}-6 \lambda ^{2} c_{1}\right ) \hypergeom \left (\left [\right ], \left [\frac {n +3}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )\right ) \left (\cos ^{2}\left (\lambda x \right )\right ) \sin \left (\lambda x \right )+\left (2 \left (\sin ^{2+n}\left (\lambda x \right )\right ) a n +6 a \left (\sin ^{2+n}\left (\lambda x \right )\right )\right ) \hypergeom \left (\left [\right ], \left [\frac {2 n +3}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right ) \left (\cos ^{2}\left (\lambda x \right )\right )}{2 \cos \left (\lambda x \right ) \left (n +1\right ) \lambda \sin \left (\lambda x \right ) \left (n +3\right ) \left (c_{1} \sin \left (\lambda x \right ) \hypergeom \left (\left [\right ], \left [\frac {n +3}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )+\hypergeom \left (\left [\right ], \left [\frac {n +1}{2+n}\right ], -\frac {a \left (\sin ^{2+n}\left (\lambda x \right )\right )}{\lambda ^{2} \left (2+n \right )^{2}}\right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2-1/2*\[Lambda]^2-3/4*\[Lambda]^2*Tan[\[Lambda]*x]^2+a*Cos[\[Lambda]*x]^2*Sin[\[Lambda]*x]^n,y[x],x,IncludeSingularSolutions -> True]
Not solved