Internal problem ID [9764]
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]
\[ \boxed {\left (\sin \left (\lambda x \right ) a +b \right ) \left (y^{\prime }-y^{2}\right )-\lambda ^{2} \sin \left (\lambda x \right ) a=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 650
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 {\lambda \left (\left (\left (-2 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) \sqrt {-a^{2}+b^{2}}\, a \,b^{3}+a^{4} b -b^{3} a^{2}\right ) \sin \left (\frac {\lambda x}{2}\right )^{2}+\left (2 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) \sqrt {-a^{2}+b^{2}}\, a \,b^{3}-a^{4} b +b^{3} a^{2}\right ) \cos \left (\frac {\lambda x}{2}\right )^{2}+2 \cos \left (\lambda x \right ) \sqrt {-a^{2}+b^{2}}\, c_{1} a b -b^{3} a^{2}+b^{5}\right ) \tan \left (\frac {\lambda x}{2}\right )^{2}+\left (\left (-4 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) \sqrt {-a^{2}+b^{2}}\, a^{2} b^{2}+2 a^{5}-2 a^{3} b^{2}\right ) \sin \left (\frac {\lambda x}{2}\right )^{2}+\left (4 a^{4} b -4 b^{3} a^{2}\right ) \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+\left (4 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) \sqrt {-a^{2}+b^{2}}\, a^{2} b^{2}-2 a^{5}+2 a^{3} b^{2}\right ) \cos \left (\frac {\lambda x}{2}\right )^{2}+4 \cos \left (\lambda x \right ) \sqrt {-a^{2}+b^{2}}\, c_{1} a^{2}\right ) \tan \left (\frac {\lambda x}{2}\right )+\left (-2 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) \sqrt {-a^{2}+b^{2}}\, a \,b^{3}+a^{4} b -b^{3} a^{2}\right ) \sin \left (\frac {\lambda x}{2}\right )^{2}+\left (2 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) \sqrt {-a^{2}+b^{2}}\, a \,b^{3}-a^{4} b +b^{3} a^{2}\right ) \cos \left (\frac {\lambda x}{2}\right )^{2}+2 \cos \left (\lambda x \right ) \sqrt {-a^{2}+b^{2}}\, c_{1} a b -b^{3} a^{2}+b^{5}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (\tan \left (\frac {\lambda x}{2}\right )^{2} b +2 \tan \left (\frac {\lambda x}{2}\right ) a +b \right ) \left (\left (2 \arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{2}+\sqrt {-a^{2}+b^{2}}\, a^{2}\right ) \cos \left (\frac {\lambda x}{2}\right ) \sin \left (\frac {\lambda x}{2}\right )+a b \sqrt {-a^{2}+b^{2}}\, \cos \left (\frac {\lambda x}{2}\right )^{2}+\arctan \left (\frac {b \tan \left (\frac {\lambda x}{2}\right )+a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{3}+\sin \left (\lambda x \right ) c_{1} a +c_{1} b \right )} \]
✓ Solution by Mathematica
Time used: 23.754 (sec). Leaf size: 186
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} (a c_1 \lambda (b-a) (a+b) \cos (\lambda x)-a \sin (\lambda x)+b)\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} (-a \cos (\lambda x)+c_1 \lambda (a-b) (a+b) (a \sin (\lambda x)+b))} \\ y(x)\to -\frac {a \lambda \cos (\lambda x)}{a \sin (\lambda x)+b} \\ \end{align*}