61.12.1 problem 38

Internal problem ID [12212]
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
Date solved : Tuesday, January 28, 2025 at 01:15:25 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 204

dsolve(diff(y(x),x)=y(x)^2+a*lambda+a*(lambda-a)*cot(lambda*x)^2,y(x), singsol=all)
 

\[ y = \frac {\left (\cos \left (\lambda x \right ) \operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) a +\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} a \cos \left (\lambda x \right )-\lambda \left (\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )\right )\right ) \csc \left (\lambda x \right )}{\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )} \]

✗ Solution by Mathematica

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

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

Not solved