Internal problem ID [9782]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-3. Equations with tangent.
Problem number: 27.
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 (\tan ^{2}\left (\lambda x \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 509
dsolve(diff(y(x),x)=y(x)^2+a*lambda+a*(lambda-a)*tan(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 }, \sin \left (\lambda x \right )\right ) c_{1} a +\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) a \right ) \sin \left (3 \lambda x \right )}{\left (\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_{1}+\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )\right ) \left (\cos \left (3 \lambda x \right )+3 \cos \left (\lambda x \right )\right )}-\frac {\left (\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_{1} a +\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) a \right ) \sin \left (\lambda x \right )}{\left (\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_{1}+\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )\right ) \left (\cos \left (3 \lambda x \right )+3 \cos \left (\lambda x \right )\right )}-\frac {\left (-2 \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_{1} \lambda -2 \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) \lambda \right ) \cos \left (2 \lambda x \right )-2 \LegendreQ \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_{1} \lambda -2 \LegendreP \left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) \lambda }{\left (\LegendreQ \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right ) c_{1}+\LegendreP \left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sin \left (\lambda x \right )\right )\right ) \left (\cos \left (3 \lambda x \right )+3 \cos \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)*Tan[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
Not solved