Internal problem ID [9839]
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: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-\lambda \arctan \relax (x )^{n} \left (y-a \,x^{m}-b \right )^{2}-a m \,x^{m -1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 50
dsolve(diff(y(x),x)=lambda*arctan(x)^n*(y(x)-a*x^m-b)^2+a*m*x^(m-1),y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (-2 x^{m} \arctan \relax (x )^{n} a \lambda -2 \arctan \relax (x )^{n} \lambda b \right ) \arctan \relax (x )^{-n}}{2 \lambda }+\frac {1}{c_{1}-\left (\int \arctan \relax (x )^{n} \lambda d x \right )} \]
✓ Solution by Mathematica
Time used: 1.332 (sec). Leaf size: 44
DSolve[y'[x]==\[Lambda]*ArcTan[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 \text {ArcTan}(K[2])^ndK[2]+c_1}+a x^m+b \\ y(x)\to a x^m+b \\ \end{align*}