61.18.1 problem 29

Internal problem ID [12262]
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
Date solved : Tuesday, January 28, 2025 at 01:52:10 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+\lambda x \operatorname {arccot}\left (x \right )^{n} y+\lambda \operatorname {arccot}\left (x \right )^{n} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 78

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

\[ y = \frac {{\mathrm e}^{\int \frac {\operatorname {arccot}\left (x \right )^{n} \lambda \,x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {\operatorname {arccot}\left (x \right )^{n} \lambda \,x^{2}-2}{x}d x}d x -c_{1}}{\left (c_{1} -\int {\mathrm e}^{\int \frac {\operatorname {arccot}\left (x \right )^{n} \lambda \,x^{2}-2}{x}d x}d x \right ) x} \]

✓ Solution by Mathematica

Time used: 2.489 (sec). Leaf size: 120

DSolve[D[y[x],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 {\exp \left (-\int _1^x-\lambda \cot ^{-1}(K[1])^n K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-\lambda \cot ^{-1}(K[1])^n K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\lambda \cot ^{-1}(K[1])^n K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}