Internal problem ID [9798]
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]
Solve \begin {gather*} \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-a \,x^{k +1} \left (\cot ^{m}\relax (x )\right ) y+a \left (\cot ^{m}\relax (x )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 273
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)
\[ y \relax (x ) = -\frac {\left ({\mathrm e}^{\int \frac {\left (\cot ^{m}\relax (x )\right ) x^{k} a \,x^{2}-2 k -2}{x}d x} x^{k} x -\left (\int \left (-x^{k} {\mathrm e}^{-2 k \left (\int \frac {1}{x}d x \right )+a \left (\int x^{k +1} \left (\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}\right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right )} k -x^{k} {\mathrm e}^{-2 k \left (\int \frac {1}{x}d x \right )+a \left (\int x^{k +1} \left (\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}\right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x \right )-c_{1}\right ) x^{-k}}{x \left (\int \left (-x^{k} {\mathrm e}^{-2 k \left (\int \frac {1}{x}d x \right )+a \left (\int x^{k +1} \left (\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}\right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right )} k -x^{k} {\mathrm e}^{-2 k \left (\int \frac {1}{x}d x \right )+a \left (\int x^{k +1} \left (\frac {i \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}\right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1}\right )} \]
✗ 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