Internal problem ID [9843]
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: 29.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda x \mathrm {arccot}\relax (x )^{n} y-\mathrm {arccot}\relax (x )^{n} \lambda =0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 55
dsolve(diff(y(x),x)=y(x)^2+lambda*x*arccot(x)^n*y(x)+lambda*arccot(x)^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {{\mathrm e}^{\int \frac {\mathrm {arccot}\relax (x )^{n} \lambda \,x^{2}-2}{x}d x}}{c_{1}-\left (\int {\mathrm e}^{\int \frac {\mathrm {arccot}\relax (x )^{n} \lambda \,x^{2}-2}{x}d x}d x \right )}-\frac {1}{x} \]
✓ Solution by Mathematica
Time used: 4.33 (sec). Leaf size: 82
DSolve[y'[x]==y[x]^2+\[Lambda]*x*ArcCot[x]^n*y[x]+\[Lambda]*ArcCot[x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x+\frac {\exp \left (-\int _1^x-\lambda \cot ^{-1}(K[5])^n K[5]dK[5]\right )}{\int _1^x\frac {\exp \left (-\int _1^{K[6]}-\lambda \cot ^{-1}(K[5])^n K[5]dK[5]\right )}{K[6]^2}dK[6]+c_1}}{x^2} \\ y(x)\to -\frac {1}{x} \\ \end{align*}