61.18.6 problem 34

Internal problem ID [12267]
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 : 34
Date solved : Tuesday, January 28, 2025 at 07:52:08 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\lambda \operatorname {arccot}\left (x \right )^{n} \left (y-a \,x^{m}-b \right )^{2}+a m \,x^{m -1} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 24

dsolve(diff(y(x),x)=lambda*arccot(x)^n*(y(x)-a*x^m-b)^2+a*m*x^(m-1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.788 (sec). Leaf size: 44

DSolve[D[y[x],x]==\[Lambda]*ArcCot[x]^n*(y[x]-a*x^m-b)^2+a*m*x^(m-1),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{-\int _1^x\lambda \cot ^{-1}(K[2])^ndK[2]+c_1}+a x^m+b \\ y(x)\to a x^m+b \\ \end{align*}