Internal problem ID [10566]
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]
\[ \boxed {y^{\prime }-y^{2}=-\frac {\lambda ^{2}}{2}-\frac {3 \tan \left (\lambda x \right )^{2} \lambda ^{2}}{4}+a \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 411
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 \left (x \right ) = -\frac {\left (c_{1} \lambda ^{2} n^{2}+4 c_{1} \lambda ^{2} n +3 c_{1} \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {n +3}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\right )^{2}}\right ) \sin \left (\lambda x \right )^{3}+\left (\lambda ^{2} n^{2}+4 \lambda ^{2} n +3 \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1+n}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\right )^{2}}\right ) \sin \left (\lambda x \right )^{2}+\left (\left (-2 \sin \left (\lambda x \right )^{n +2} c_{1} a n -2 \sin \left (\lambda x \right )^{n +2} c_{1} a \right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2 n +5}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\right )^{2}}\right )+\left (2 c_{1} \lambda ^{2} n^{2}+8 c_{1} \lambda ^{2} n +6 c_{1} \lambda ^{2}\right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {n +3}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\right )^{2}}\right )\right ) \cos \left (\lambda x \right )^{2} \sin \left (\lambda x \right )+\left (-2 \sin \left (\lambda x \right )^{n +2} a n -6 \sin \left (\lambda x \right )^{n +2} a \right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3+2 n}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\right )^{2}}\right ) \cos \left (\lambda x \right )^{2}}{2 \cos \left (\lambda x \right ) \left (1+n \right ) \lambda \sin \left (\lambda x \right ) \left (n +3\right ) \left (c_{1} \sin \left (\lambda x \right ) \operatorname {hypergeom}\left (\left [\right ], \left [\frac {n +3}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\right )^{2}}\right )+\operatorname {hypergeom}\left (\left [\right ], \left [\frac {1+n}{n +2}\right ], -\frac {\sin \left (\lambda x \right )^{n +2} a}{\lambda ^{2} \left (n +2\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