Internal problem ID [10555]
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: 57.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=2 a b +\lambda a +b \lambda +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}+b \left (\lambda -b \right ) \cot \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 268
dsolve(diff(y(x),x)=y(x)^2+lambda*a+lambda*b+2*a*b+a*(lambda-a)*tan(lambda*x)^2+b*(lambda-b)*cot(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {4 c_{1} \lambda \cos \left (x \lambda \right )^{2} \sin \left (x \lambda \right )^{2} \left (b -\lambda +a \right ) \operatorname {hypergeom}\left (\left [2, \frac {2 \lambda -b -a}{\lambda }\right ], \left [-\frac {2 a -5 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )-2 c_{1} \left (\left (-3 \lambda ^{2}+\left (\frac {7 a}{2}+\frac {3 b}{2}\right ) \lambda -a b \right ) \cos \left (x \lambda \right )^{2}+a^{2} \sin \left (x \lambda \right )^{2}-\frac {5 \left (a -\frac {3 \lambda }{5}\right ) \lambda }{2}\right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {2 a -3 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )+2 \left (a -\frac {3 \lambda }{2}\right ) \sin \left (x \lambda \right )^{\frac {2 b}{\lambda }} \left (a \tan \left (x \lambda \right )-b \cot \left (x \lambda \right )\right ) \cos \left (x \lambda \right )^{\frac {2 a}{\lambda }}}{\left (2 a -3 \lambda \right ) \left (c_{1} \cos \left (x \lambda \right ) \sin \left (x \lambda \right ) \operatorname {hypergeom}\left (\left [1, \frac {-b +\lambda -a}{\lambda }\right ], \left [-\frac {2 a -3 \lambda }{2 \lambda }\right ], \cos \left (x \lambda \right )^{2}\right )+\cos \left (x \lambda \right )^{\frac {2 a}{\lambda }} \sin \left (x \lambda \right )^{\frac {2 b}{\lambda }}\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2+\[Lambda]*a+\[Lambda]*b+2*a*b+a*(\[Lambda]-a)*Tan[\[Lambda]*x]^2+b*(\[Lambda]-b)*Cot[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved