Internal problem ID [10535]
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]
\[ \boxed {y^{\prime }-y^{2}=\lambda a +a \left (\lambda -a \right ) \tan \left (\lambda x \right )^{2}} \]
✓ 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: 2.982 (sec). Leaf size: 259
DSolve[y'[x]==y[x]^2+a*\[Lambda]+a*(\[Lambda]-a)*Tan[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \left (a c_1 \sin ^2(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )+(2 a-\lambda ) \sqrt {\sin ^2(\lambda x)} \left (a \sin (\lambda x) \cos ^{\frac {2 a}{\lambda }-1}(\lambda x)-c_1\right )\right )}{2 (2 a-\lambda ) \sqrt {\sin ^2(\lambda x)} \cos ^{\frac {2 a}{\lambda }}(\lambda x)+c_1 \sin (2 \lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )} y(x)\to \frac {\tan (\lambda x) \left (a \sqrt {\sin ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )-2 a+\lambda \right )}{\sqrt {\sin ^2(\lambda x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-\frac {a}{\lambda },\frac {3}{2}-\frac {a}{\lambda },\cos ^2(x \lambda )\right )} \end{align*}