Internal problem ID [10547]
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: 39.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}=3 \lambda a +\lambda ^{2}+a \left (\lambda -a \right ) \cot \left (\lambda x \right )^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 436
dsolve(diff(y(x),x)=y(x)^2+lambda^2+3*a*lambda+a*(lambda-a)*cot(lambda*x)^2,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (\left (2 c_{1} a +3 c_{1} \lambda \right ) \operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )+\left (2 a +3 \lambda \right ) \operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right )\right ) \cos \left (\lambda x \right )-4 \operatorname {LegendreQ}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} \lambda -4 \operatorname {LegendreP}\left (\frac {2 a +3 \lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda \right ) \sin \left (\lambda x \right )}{2 \left (\cos \left (\lambda x \right )^{2}-1\right ) \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 )}-\frac {\left (\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} \lambda +\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda \right ) \cos \left (\lambda x \right )^{3}+\left (-\operatorname {LegendreQ}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) c_{1} \lambda -\operatorname {LegendreP}\left (\frac {2 a +\lambda }{2 \lambda }, \frac {2 a -\lambda }{2 \lambda }, \cos \left (\lambda x \right )\right ) \lambda \right ) \cos \left (\lambda x \right )}{2 \left (\cos \left (\lambda x \right )^{2}-1\right ) \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 ) \sin \left (\lambda x \right )} \]
✓ Solution by Mathematica
Time used: 67.099 (sec). Leaf size: 306
DSolve[y'[x]==y[x]^2+\[Lambda]^2+3*a*\[Lambda]+a*(\[Lambda]-a)*Cot[\[Lambda]*x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda x) e^{-\text {arctanh}(\cos (2 \lambda x))} \left (c_1 \sin ^{\frac {a}{\lambda }}(2 \lambda x) ((a+\lambda ) \cos (2 \lambda x)+a-\lambda ) e^{\text {arctanh}(\cos (2 \lambda x))} \int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]+\sin ^{\frac {a}{\lambda }}(2 \lambda x) ((a+\lambda ) \cos (2 \lambda x)+a-\lambda ) e^{\text {arctanh}(\cos (2 \lambda x))}+c_1 e^{\frac {a \text {arctanh}(\cos (2 \lambda x))}{\lambda }}\right )}{1+c_1 \int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]} y(x)\to \csc (2 \lambda x) \left (-\frac {\sin ^{-\frac {a}{\lambda }}(2 \lambda x) e^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda x))}{\lambda }}}{\int _1^xe^{\frac {(a-\lambda ) \text {arctanh}(\cos (2 \lambda K[1]))}{\lambda }} \sin ^{-\frac {a+\lambda }{\lambda }}(2 \lambda K[1])dK[1]}-(a+\lambda ) \cos (2 \lambda x)-a+\lambda \right ) \end{align*}