61.12.8 problem 45

Internal problem ID [12219]
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 : 45
Date solved : Tuesday, January 28, 2025 at 01:19:25 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.105 (sec). Leaf size: 31

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

\[ y = -\tan \left (-a b \left (\int x^{k -1} \cot \left (\lambda x \right )^{m}d x \right )+c_{1} \right ) b \,x^{k} \]

✓ Solution by Mathematica

Time used: 0.556 (sec). Leaf size: 50

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

\[ y(x)\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^xa \cot ^m(\lambda K[1]) K[1]^{k-1}dK[1]+c_1\right ) \]