61.18.3 problem 31

Internal problem ID [12264]
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 : 31
Date solved : Tuesday, January 28, 2025 at 07:51:56 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 87

dsolve(diff(y(x),x)=lambda*arccot(x)^n*y(x)^2+a*y(x)+a*b-b^2*lambda*arccot(x)^n,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.942 (sec). Leaf size: 240

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

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