Internal problem ID [9842]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-4. Equations containing
arccotangent.
Problem number: 28.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\lambda \mathrm {arccot}\relax (x )^{n} y+a^{2}-a \lambda \mathrm {arccot}\relax (x )^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 97
dsolve(diff(y(x),x)=y(x)^2+lambda*arccot(x)^n*y(x)-a^2+a*lambda*arccot(x)^n,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\int {\mathrm e}^{\int \left (\mathrm {arccot}\relax (x )^{n} \lambda -2 a \right )d x}d x \right ) {\mathrm e}^{\int \left (-\mathrm {arccot}\relax (x )^{n} \lambda +2 a \right )d x} a +c_{1} {\mathrm e}^{\int \left (-\mathrm {arccot}\relax (x )^{n} \lambda +2 a \right )d x} a +1\right ) {\mathrm e}^{\int \left (\mathrm {arccot}\relax (x )^{n} \lambda -2 a \right )d x}}{c_{1}+\int {\mathrm e}^{\int \left (\mathrm {arccot}\relax (x )^{n} \lambda -2 a \right )d x}d x} \]
✓ Solution by Mathematica
Time used: 5.021 (sec). Leaf size: 210
DSolve[y'[x]==y[x]^2+\[Lambda]*ArcCot[x]^n*y[x]-a^2+a*\[Lambda]*ArcCot[x]^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[6]}\left (2 a-\lambda \cot ^{-1}(K[5])^n\right )dK[5]\right ) \left (-\lambda \cot ^{-1}(K[6])^n+a-y(x)\right )}{n \lambda (a+y(x))}dK[6]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[6]}\left (2 a-\lambda \cot ^{-1}(K[5])^n\right )dK[5]\right ) \left (-\lambda \cot ^{-1}(K[6])^n+a-K[7]\right )}{n \lambda (a+K[7])^2}+\frac {\exp \left (-\int _1^{K[6]}\left (2 a-\lambda \cot ^{-1}(K[5])^n\right )dK[5]\right )}{n \lambda (a+K[7])}\right )dK[6]-\frac {\exp \left (-\int _1^x\left (2 a-\lambda \cot ^{-1}(K[5])^n\right )dK[5]\right )}{n \lambda (a+K[7])^2}\right )dK[7]=c_1,y(x)\right ] \]