Internal problem ID [9795]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}+2 a b \cot \left (x a \right ) y-b^{2}+a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 661
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 \relax (x ) = \frac {\left (\left (\left (-2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}\, c_{1}-c_{1} a \right ) \LegendreQ \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (-2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}-a \right ) \LegendreP \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \cos \left (a x \right )+\left (-2 c_{1} b a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}\, c_{1}+2 c_{1} a \right ) \LegendreQ \left (\frac {a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (-2 b a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}+2 a \right ) \LegendreP \left (\frac {a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \sin \left (a x \right )}{2 \left (\cos ^{2}\left (a x \right )-1\right ) \left (\LegendreQ \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1}+\LegendreP \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right )}+\frac {\left (\left (2 c_{1} b a -c_{1} a \right ) \LegendreQ \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (2 b a -a \right ) \LegendreP \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \left (\cos ^{3}\left (a x \right )\right )+\left (\left (-2 c_{1} b a +c_{1} a \right ) \LegendreQ \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (-2 b a +a \right ) \LegendreP \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \cos \left (a x \right )}{2 \left (\cos ^{2}\left (a x \right )-1\right ) \left (\LegendreQ \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1}+\LegendreP \left (\frac {-a +2 \sqrt {b^{2} a^{2}-a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\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