Internal problem ID [9849]
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: 35.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }-\lambda \mathrm {arccot}\relax (x )^{n} y^{2}-k y-\lambda \,b^{2} x^{2 k} \mathrm {arccot}\relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 34
dsolve(x*diff(y(x),x)=lambda*arccot(x)^n*y(x)^2+k*y(x)+lambda*b^2*x^(2*k)*arccot(x)^n,y(x), singsol=all)
\[ y \relax (x ) = -\tan \left (-b \lambda \left (\int \frac {x^{k} \mathrm {arccot}\relax (x )^{n}}{x}d x \right )+c_{1}\right ) b \,x^{k} \]
✓ Solution by Mathematica
Time used: 1.608 (sec). Leaf size: 48
DSolve[x*y'[x]==\[Lambda]*ArcCot[x]^n*y[x]^2+k*y[x]+\[Lambda]*b^2*x^(2*k)*ArcCot[x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {b^2} x^k \tan \left (\sqrt {b^2} \int _1^x\lambda \cot ^{-1}(K[1])^n K[1]^{k-1}dK[1]+c_1\right ) \\ \end{align*}