Internal problem ID [10538]
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]
\[ \boxed {y^{\prime }-y^{2}+2 a b \cot \left (x a \right ) y=-a^{2}+b^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 291
dsolve(diff(y(x),x)=y(x)^2-2*a*b*cot(a*x)*y(x)+b^2-a^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\cos \left (a x \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+c_{1} \cos \left (a x \right ) \left (a b +\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}\right ) \operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )+\left (\operatorname {LegendreQ}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )\right ) \left (-\sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}+\left (b -1\right ) a \right )\right ) \csc \left (a x \right )}{\operatorname {LegendreQ}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {-a +2 \sqrt {\left (b^{2}-1\right ) a^{2}+b^{2}}}{2 a}, b -\frac {1}{2}, \cos \left (a x \right )\right )} \]
✗ 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