Internal problem ID [10548]
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: 40.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}+2 a b \cot \left (a x \right ) y=-a^{2}+b^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 565
dsolve(diff(y(x),x)=y(x)^2-2*a*b*cot(a*x)*y(x)+b^2-a^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {\left (b a c_{1} -\sqrt {b^{2} a^{2}-a^{2}+b^{2}}\, c_{1} -c_{1} a \right ) \operatorname {LegendreQ}\left (\frac {a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right )}{\sqrt {-\sin \left (a x \right )^{2}+1}\, \left (\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right ) c_{1} +\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right )\right )}+\frac {\left (\sqrt {-\sin \left (a x \right )^{2}+1}\, c_{1} a b +\sqrt {-\sin \left (a x \right )^{2}+1}\, \sqrt {b^{2} a^{2}-a^{2}+b^{2}}\, c_{1} \right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right )+\left (b a -\sqrt {b^{2} a^{2}-a^{2}+b^{2}}-a \right ) \operatorname {LegendreP}\left (\frac {a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right )+\left (\sqrt {-\sin \left (a x \right )^{2}+1}\, a b +\sqrt {-\sin \left (a x \right )^{2}+1}\, \sqrt {b^{2} a^{2}-a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right )}{\sqrt {-\sin \left (a x \right )^{2}+1}\, \left (\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right ) c_{1} +\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \sqrt {-\sin \left (a x \right )^{2}+1}\right )\right )}\right ) \cos \left (a x \right )}{\sin \left (a x \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2-2*a*b*Cot[a*x]*y[x]+b^2-a^2,y[x],x,IncludeSingularSolutions -> True]
Not solved