Internal problem ID [9836]
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: 22.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \arctan \relax (x )^{n} y^{2}-a y-a b +b^{2} \lambda \arctan \relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 114
dsolve(diff(y(x),x)=lambda*arctan(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arctan(x)^n,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int \arctan \relax (x )^{n} \lambda \,{\mathrm e}^{\int \left (-2 \arctan \relax (x )^{n} \lambda b +a \right )d x}d x \right ) {\mathrm e}^{\int \left (2 \arctan \relax (x )^{n} \lambda b -a \right )d x} b +c_{1} {\mathrm e}^{\int \left (2 \arctan \relax (x )^{n} \lambda b -a \right )d x} b +1\right ) {\mathrm e}^{\int \left (-2 \arctan \relax (x )^{n} \lambda b +a \right )d x}}{c_{1}+\int \arctan \relax (x )^{n} \lambda \,{\mathrm e}^{\int \left (-2 \arctan \relax (x )^{n} \lambda b +a \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 6.495 (sec). Leaf size: 240
DSolve[y'[x]==\[Lambda]*ArcTan[x]^n*y[x]^2+a*y[x]+a*b-b^2*\[Lambda]*ArcTan[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 b \lambda \text {ArcTan}(K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \text {ArcTan}(K[2])^n+\lambda y(x) \text {ArcTan}(K[2])^n+a\right )}{a n \lambda (b+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\left (2 b \lambda \text {ArcTan}(K[1])^n-a\right )dK[1]\right ) \text {ArcTan}(K[2])^n}{a n (b+K[3])}-\frac {\exp \left (-\int _1^{K[2]}\left (2 b \lambda \text {ArcTan}(K[1])^n-a\right )dK[1]\right ) \left (-b \lambda \text {ArcTan}(K[2])^n+\lambda K[3] \text {ArcTan}(K[2])^n+a\right )}{a n \lambda (b+K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x\left (2 b \lambda \text {ArcTan}(K[1])^n-a\right )dK[1]\right )}{a n \lambda (b+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]