Internal problem ID [9778]
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]
\[ \boxed {y^{\prime }-y^{2}-\lambda a -a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 380
dsolve(diff(y(x),x)=y(x)^2+a*lambda+a*(lambda-a)*tan(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {c_{1} \lambda \operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right )}{\sqrt {-\cos \left (\lambda x \right )^{2}+1}\, \left (\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right )\right )}+\frac {-\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right ) \sqrt {-\cos \left (\lambda x \right )^{2}+1}\, c_{1} a -\sqrt {-\cos \left (\lambda x \right )^{2}+1}\, \operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right ) a +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right ) \lambda }{\sqrt {-\cos \left (\lambda x \right )^{2}+1}\, \left (\operatorname {LegendreQ}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right ) c_{1} +\operatorname {LegendreP}\left (\frac {2 a -\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \sqrt {-\cos \left (\lambda x \right )^{2}+1}\right )\right )}\right ) \sin \left (\lambda x \right )}{\cos \left (\lambda x \right )} \]
✗ 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