Internal problem ID [10505]
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: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {2 y^{\prime }-\left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}=-a \sin \left (\lambda x \right )-a +\lambda } \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 220
dsolve(2*diff(y(x),x)=(lambda+a-a*sin(lambda*x))*y(x)^2+lambda-a-a*sin(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (\left (\int _{}^{\sin \left (x \lambda \right )}\frac {\left (a \left (\textit {\_a} -1\right )-\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right ) c_{1} +1\right ) \sqrt {-\cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )^{2}}\, \left (a \cos \left (x \lambda \right )+\tan \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) \lambda \right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right )+\frac {\sec \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )^{2} \csc \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right ) \cos \left (x \lambda \right ) {\mathrm e}^{\frac {a \sin \left (x \lambda \right )}{\lambda }} c_{1} \lambda \left (-\lambda -a +a \sin \left (x \lambda \right )\right )}{2}\right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )\right )}{\sqrt {-\cos \left (\frac {\pi }{4}+\frac {x \lambda }{2}\right )^{2}}\, \left (\left (\int _{}^{\sin \left (x \lambda \right )}\frac {\left (a \left (\textit {\_a} -1\right )-\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{\frac {3}{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right ) c_{1} +1\right ) \left (-\lambda -a +a \sin \left (x \lambda \right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[2*y'[x]==(\[Lambda]+a-a*Sin[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Sin[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
Not solved