61.12.5 problem 42

Internal problem ID [12216]
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
Date solved : Tuesday, January 28, 2025 at 01:15:52 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+a x \cot \left (b x \right )^{m} y+a \cot \left (b x \right )^{m} \end{align*}

✓ Solution by Maple

Time used: 0.008 (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)
 

\[ y = \frac {-{\mathrm e}^{\int \frac {a \cot \left (b x \right )^{m} x^{2}-2}{x}d x} x -\int {\mathrm e}^{\int \frac {a \cot \left (b x \right )^{m} x^{2}-2}{x}d x}d x +c_{1}}{\left (-c_{1} +\int {\mathrm e}^{\int \frac {a \cot \left (b x \right )^{m} x^{2}-2}{x}d x}d x \right ) x} \]

✓ Solution by Mathematica

Time used: 2.671 (sec). Leaf size: 126

DSolve[D[y[x],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*}