Internal problem ID [10549]
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: 41.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a \cot \left (\beta x \right ) y=a b \cot \left (\beta x \right )-b^{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 61
dsolve(diff(y(x),x)=y(x)^2+a*cot(beta*x)*y(x)+a*b*cot(beta*x)-b^2,y(x), singsol=all)
\[ y \left (x \right ) = -b -\frac {\left (\cot \left (\beta x \right )^{2}+1\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 x b}}{\int \left (\cot \left (\beta x \right )^{2}+1\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 x b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 26.26 (sec). Leaf size: 462
DSolve[y'[x]==y[x]^2+a*Cot[\[Beta]*x]*y[x]+a*b*Cot[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b (i a+2 b-2 i \beta ) \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta } \operatorname {Hypergeometric2F1}\left (1,\frac {a+2 i b}{2 \beta },-\frac {a-2 i b-2 \beta }{2 \beta },e^{2 i x \beta }\right )+(a-2 i b) \left ((a-2 i b-2 \beta ) \left (\left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta }-a b \beta c_1 2^{a/\beta } e^{2 b x}\right )-i b e^{2 i \beta x} \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta } \operatorname {Hypergeometric2F1}\left (1,\frac {a+2 i b+2 \beta }{2 \beta },-\frac {a-2 i b-4 \beta }{2 \beta },e^{2 i x \beta }\right )\right )}{i (-a+2 i b+2 \beta ) \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta } \operatorname {Hypergeometric2F1}\left (1,\frac {a+2 i b}{2 \beta },-\frac {a-2 i b-2 \beta }{2 \beta },e^{2 i x \beta }\right )+(a-2 i b) \left (i e^{2 i \beta x} \left (-i e^{-i \beta x} \left (-1+e^{2 i \beta x}\right )\right )^{a/\beta } \operatorname {Hypergeometric2F1}\left (1,\frac {a+2 i b+2 \beta }{2 \beta },-\frac {a-2 i b-4 \beta }{2 \beta },e^{2 i x \beta }\right )+a \beta c_1 2^{a/\beta } e^{2 b x} (a-2 i b-2 \beta )\right )} y(x)\to -b \end{align*}