61.16.1 problem 19

Internal problem ID [12252]
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
Date solved : Tuesday, January 28, 2025 at 01:45:06 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 71

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)
 

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

✓ Solution by Mathematica

Time used: 1.104 (sec). Leaf size: 210

DSolve[D[y[x],x]==y[x]^2+\[Lambda]*ArcTan[x]^n*y[x]-a^2+a*\[Lambda]*ArcTan[x]^n,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right ) \left (-\lambda \arctan (K[2])^n+a-y(x)\right )}{n \lambda (a+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right )}{n \lambda (a+K[3])^2}-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right ) \left (-\lambda \arctan (K[2])^n+a-K[3]\right )}{n \lambda (a+K[3])^2}-\frac {\exp \left (-\int _1^{K[2]}\left (2 a-\lambda \arctan (K[1])^n\right )dK[1]\right )}{n \lambda (a+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]