Internal problem ID [9793]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-4. Equations with cotangent.
Problem number: 38.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda a -a \left (\lambda -a \right ) \left (\cot ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 353
dsolve(diff(y(x),x)=y(x)^2+a*lambda+a*(lambda-a)*cot(lambda*x)^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} a +\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) a \right ) \cos \left (3 \lambda x \right )+\left (-2 \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} \lambda -2 \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda \right ) \cos \left (2 \lambda x \right )+\left (-\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} a -\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) a \right ) \cos \left (\lambda x \right )+2 \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} \lambda +2 \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda }{\left (\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1}+\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )\right ) \left (\sin \left (3 \lambda x \right )-3 \sin \left (\lambda x \right )\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2+a*\[Lambda]+a*(\[Lambda]-a)*Cot[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved