Internal problem ID [10541]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-a \,x^{k +1} \cot \left (x \right )^{m} y=-a \cot \left (x \right )^{m}} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 170
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)
\[ y \left (x \right ) = \frac {x^{-1-k} \left (x^{1+k} {\mathrm e}^{\int \frac {x^{1+k} \cot \left (x \right )^{m} a x -2 k -2}{x}d x}+\left (\int x^{k} {\mathrm e}^{\int \frac {x^{1+k} \cot \left (x \right )^{m} a x -2 k -2}{x}d x}d x \right ) k +\int x^{k} {\mathrm e}^{\int \frac {x^{1+k} \cot \left (x \right )^{m} a x -2 k -2}{x}d x}d x +c_{1} \right )}{\left (\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \cot \left (x \right )^{m}-2 k -2}{x}d x}d x \right ) k +c_{1} +\int x^{k} {\mathrm e}^{\int \frac {a \,x^{k +2} \cot \left (x \right )^{m}-2 k -2}{x}d x}d x} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[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