Internal problem ID [9796]
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]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a \cot \left (\beta x \right ) y-a b \cot \left (\beta x \right )+b^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \relax (x ) = -b -\frac {\left (\cot ^{2}\left (\beta x \right )+1\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}}{\int \left (\cot ^{2}\left (\beta x \right )+1\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 b x}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 14.337 (sec). Leaf size: 296
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 {\left (2-2 e^{2 i \beta x}\right )^{a/\beta } \left ((a+i b) (a-2 i b) \sin ^{\frac {a}{\beta }}(\beta x)-a b \beta c_1 \left (a^2+4 b^2\right ) e^{2 b x}\right )+i a b 2^{\frac {a+\beta }{\beta }} \sin ^{\frac {a}{\beta }}(\beta x) \, _2F_1\left (1-\frac {a}{\beta },-\frac {a-2 i b}{2 \beta };1-\frac {a-2 i b}{2 \beta };e^{2 i x \beta }\right )}{a \beta c_1 \left (a^2+4 b^2\right ) e^{2 b x} \left (2-2 e^{2 i \beta x}\right )^{a/\beta }+(2 b+i a) \sin ^{\frac {a}{\beta }}(\beta x) \left (\frac {a 2^{a/\beta } \left (e^{2 i \beta x}\right )^{\frac {a-2 i b}{2 \beta }} B_{e^{2 i x \beta }}\left (-\frac {a-2 i b}{2 \beta },\frac {a}{\beta }\right )}{\beta }+\left (2-2 e^{2 i \beta x}\right )^{a/\beta }\right )} \\ y(x)\to -b \\ \end{align*}