Internal problem ID [10586]
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: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2}-\lambda \arctan \left (x \right )^{n} y=-a^{2}+a \lambda \arctan \left (x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 97
dsolve(diff(y(x),x)=y(x)^2+lambda*arctan(x)^n*y(x)-a^2+a*lambda*arctan(x)^n,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (\int {\mathrm e}^{\int \left (\arctan \left (x \right )^{n} \lambda -2 a \right )d x}d x \right ) {\mathrm e}^{\int \left (-\arctan \left (x \right )^{n} \lambda +2 a \right )d x} a +c_{1} {\mathrm e}^{\int \left (-\arctan \left (x \right )^{n} \lambda +2 a \right )d x} a +1\right ) {\mathrm e}^{\int \left (\arctan \left (x \right )^{n} \lambda -2 a \right )d x}}{c_{1} +\int {\mathrm e}^{\int \left (\arctan \left (x \right )^{n} \lambda -2 a \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 7.862 (sec). Leaf size: 210
DSolve[y'[x]==y[x]^2+\[Lambda]*ArcTan[x]^n*y[x]-a^2+a*\[Lambda]*ArcTan[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right ) \left (-\lambda \arctan (K[2])^n+a-y(x)\right )}{n \lambda (a+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right )}{n \lambda (a+K[3])^2}-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right ) \left (-\lambda \arctan (K[2])^n+a-K[3]\right )}{n \lambda (a+K[3])^2}-\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right )}{n \lambda (a+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]