61.9.7 problem 7

Internal problem ID [12181]
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
Date solved : Tuesday, January 28, 2025 at 12:52:00 AM
CAS classification : [_Riccati]

\begin{align*} 2 y^{\prime }&=\left (\lambda +a -a \sin \left (\lambda x \right )\right ) y^{2}+\lambda -a -a \sin \left (\lambda x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (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)
 

\[ y = -\frac {\operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right ) \left (\left (a \cos \left (\lambda x \right )+\tan \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \lambda \right ) \sqrt {-\cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}\, \left (\left (\int _{}^{\sin \left (\lambda x \right )}\frac {\left (a \left (\textit {\_a} -1\right )-\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{{3}/{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right ) c_{1} +1\right ) \operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )\right )+\frac {\sec \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2} \csc \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right ) \cos \left (\lambda x \right ) {\mathrm e}^{\frac {a \sin \left (\lambda x \right )}{\lambda }} c_{1} \lambda \left (-\lambda -a +a \sin \left (\lambda x \right )\right )}{2}\right )}{\sqrt {-\cos \left (\frac {\pi }{4}+\frac {\lambda x}{2}\right )^{2}}\, \left (\left (\int _{}^{\sin \left (\lambda x \right )}\frac {\left (a \left (\textit {\_a} -1\right )-\lambda \right ) {\mathrm e}^{\frac {a \textit {\_a}}{\lambda }}}{\left (\textit {\_a} -1\right )^{{3}/{2}} \sqrt {\textit {\_a} +1}}d \textit {\_a} \right ) c_{1} +1\right ) \left (-\lambda -a +a \sin \left (\lambda x \right )\right )} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[2*D[y[x],x]==(\[Lambda]+a-a*Sin[\[Lambda]*x])*y[x]^2+\[Lambda]-a-a*Sin[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
 

Not solved