Internal problem ID [10540]
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]
\[ \boxed {y^{\prime }-y^{2}-a x \cot \left (b x \right )^{m} y=a \cot \left (b x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = \frac {-{\mathrm e}^{\int \frac {a \cot \left (b x \right )^{m} x^{2}-2}{x}d x} x -\left (\int {\mathrm e}^{\int \frac {a \cot \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right )+c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \cot \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]
✓ Solution by Mathematica
Time used: 8.36 (sec). Leaf size: 126
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 {\exp \left (-\int _1^x-a \cot ^m(b K[1]) K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-a \cot ^m(b K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-a \cot ^m(b K[1]) K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}