61.12.6 problem 43

Internal problem ID [12217]
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 : 43
Date solved : Tuesday, January 28, 2025 at 01:15:59 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} \cot \left (x \right )^{m} y-a \cot \left (x \right )^{m} \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 174

dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+a*x^(k+1)*cot(x)^m*y(x)-a*cot(x)^m,y(x), singsol=all)
 

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

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],x]==-(k+1)*x^k*y[x]^2+a*x^(k+1)*Cot[x]^m*y[x]-a*Cot[x]^m,y[x],x,IncludeSingularSolutions -> True]
 

Not solved