Internal problem ID [9792]
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: 41.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-a \cot \left (\beta x \right ) y-a b \cot \left (\beta x \right )+b^{2}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 61
dsolve(diff(y(x),x)=y(x)^2+a*cot(beta*x)*y(x)+a*b*cot(beta*x)-b^2,y(x), singsol=all)
\[ y \left (x \right ) = -b -\frac {\left (\cot \left (\beta x \right )^{2}+1\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 x b}}{\int \left (\cot \left (\beta x \right )^{2}+1\right )^{-\frac {a}{2 \beta }} {\mathrm e}^{-2 x b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 82.06 (sec). Leaf size: 19223
DSolve[y'[x]==y[x]^2+a*Cot[\[Beta]*x]*y[x]+a*b*Cot[\[Beta]*x]-b^2,y[x],x,IncludeSingularSolutions -> True]
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