Internal problem ID [10511]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-1. Equations with
sine
Problem number: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (a \sin \left (\lambda x \right )+b \right ) \left (y^{\prime }-y^{2}\right )=a \,\lambda ^{2} \sin \left (\lambda x \right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 261
dsolve((a*sin(lambda*x)+b)*(diff(y(x),x)-y(x)^2)-a*lambda^2*sin(lambda*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-2 \left (a -b \right ) b^{2} \left (a +b \right ) \left (\cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {1}{2}\right ) a \sqrt {-a^{2}+b^{2}}\, \arctan \left (\frac {\tan \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right )+2 c_{1} \left (\cos \left (\frac {x \lambda }{2}\right )^{2}-\frac {1}{2}\right ) a \sqrt {-a^{2}+b^{2}}+\left (a -b \right )^{2} \left (a^{2} \cos \left (\frac {x \lambda }{2}\right )^{2}-\sin \left (\frac {x \lambda }{2}\right ) a b \cos \left (\frac {x \lambda }{2}\right )-\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \left (a +b \right )^{2}\right ) \lambda }{\sqrt {-a^{2}+b^{2}}\, \left (2 \left (a -b \right ) b^{2} \left (a +b \right ) \left (\sin \left (\frac {x \lambda }{2}\right ) a \cos \left (\frac {x \lambda }{2}\right )+\frac {b}{2}\right ) \arctan \left (\frac {\tan \left (\frac {x \lambda }{2}\right ) b +a}{\sqrt {-a^{2}+b^{2}}}\right )+a \cos \left (\frac {x \lambda }{2}\right ) \left (a -b \right ) \left (a +b \right ) \left (a \sin \left (\frac {x \lambda }{2}\right )+b \cos \left (\frac {x \lambda }{2}\right )\right ) \sqrt {-a^{2}+b^{2}}-2 c_{1} \left (\sin \left (\frac {x \lambda }{2}\right ) a \cos \left (\frac {x \lambda }{2}\right )+\frac {b}{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 24.795 (sec). Leaf size: 189
DSolve[(a*Sin[\[Lambda]*x]+b)*(y'[x]-y[x]^2)-a*\[Lambda]^2*Sin[\[Lambda]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\lambda \left (2 a b \cos (\lambda x) \arctan \left (\frac {a+b \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {b^2-a^2}}\right )+\sqrt {b^2-a^2} \left (-a c_1 \lambda \left (a^2-b^2\right ) \cos (\lambda x)-a \sin (\lambda x)+b\right )\right )}{-2 b (a \sin (\lambda x)+b) \arctan \left (\frac {a+b \tan \left (\frac {\lambda x}{2}\right )}{\sqrt {b^2-a^2}}\right )+\sqrt {b^2-a^2} \left (-a \cos (\lambda x)+c_1 \lambda \left (a^2-b^2\right ) (a \sin (\lambda x)+b)\right )} \\ y(x)\to -\frac {a \lambda \cos (\lambda x)}{a \sin (\lambda x)+b} \\ \end{align*}