61.16.8 problem 26

Internal problem ID [12259]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-3. Equations containing arctangent.
Problem number : 26
Date solved : Tuesday, January 28, 2025 at 01:49:45 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.133 (sec). Leaf size: 29

dsolve(x*diff(y(x),x)=lambda*arctan(x)^n*y(x)^2+k*y(x)+lambda*b^2*x^(2*k)*arctan(x)^n,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.548 (sec). Leaf size: 48

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

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