Internal problem ID [9797]
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: 42.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a x \left (\cot ^{m}\left (x b \right )\right ) y-a \left (\cot ^{m}\left (x b \right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 85
dsolve(diff(y(x),x)=y(x)^2+a*x*cot(b*x)^m*y(x)+a*cot(b*x)^m,y(x), singsol=all)
\[ y \relax (x ) = -\frac {{\mathrm e}^{\int \frac {a \left (\cot ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {a \left (\cot ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x}d x -c_{1}}{\left (-c_{1}+\int {\mathrm e}^{\int \frac {a \left (\cot ^{m}\left (b x \right )\right ) x^{2}-2}{x}d x}d x \right ) x} \]
✓ Solution by Mathematica
Time used: 4.833 (sec). Leaf size: 86
DSolve[y'[x]==y[x]^2+a*x*Cot[b*x]^m*y[x]+a*Cot[b*x]^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x+\frac {\exp \left (-\int _1^x-a \cot ^m(b K[5]) K[5]dK[5]\right )}{\int _1^x\frac {\exp \left (-\int _1^{K[6]}-a \cot ^m(b K[5]) K[5]dK[5]\right )}{K[6]^2}dK[6]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}