Internal problem ID [10577]
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: 20.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-\lambda x \arctan \left (x \right )^{n} y=\arctan \left (x \right )^{n} \lambda } \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 78
dsolve(diff(y(x),x)=y(x)^2+lambda*x*arctan(x)^n*y(x)+lambda*arctan(x)^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\int \frac {\arctan \left (x \right )^{n} \lambda \,x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {\arctan \left (x \right )^{n} \lambda \,x^{2}-2}{x}d x}d x -c_{1}}{\left (c_{1} -\left (\int {\mathrm e}^{\int \frac {\arctan \left (x \right )^{n} \lambda \,x^{2}-2}{x}d x}d x \right )\right ) x} \]
✓ Solution by Mathematica
Time used: 7.063 (sec). Leaf size: 120
DSolve[y'[x]==y[x]^2+\[Lambda]*x*ArcTan[x]^n*y[x]+\[Lambda]*ArcTan[x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-\lambda \arctan (K[1])^n K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-\lambda \arctan (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 \arctan (K[1])^n K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}