Internal problem ID [10583]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime } x -\lambda \arctan \left (x \right )^{n} y^{2}-k y=\lambda \,b^{2} x^{2 k} \arctan \left (x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.031 (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)
\[ y \left (x \right ) = -\tan \left (-\lambda b \left (\int x^{-1+k} \arctan \left (x \right )^{n}d x \right )+c_{1} \right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 1.992 (sec). Leaf size: 48
DSolve[x*y'[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]
\[ y(x)\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^x\lambda \arctan (K[1])^n K[1]^{k-1}dK[1]+c_1\right ) \]