55.15.2 problem 11

Internal problem ID [13464]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-2. Equations containing arccosine.
Problem number : 11
Date solved : Wednesday, October 01, 2025 at 02:19:24 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+\lambda x \arccos \left (x \right )^{n} y+\lambda \arccos \left (x \right )^{n} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 55

ode:=diff(y(x),x) = y(x)^2+lambda*x*arccos(x)^n*y(x)+lambda*arccos(x)^n; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\int \frac {\lambda \arccos \left (x \right )^{n} x^{2}-2}{x}d x}}{c_1 -\int {\mathrm e}^{\int \frac {\lambda \arccos \left (x \right )^{n} x^{2}-2}{x}d x}d x}-\frac {1}{x} \]

✓ Mathematica. Time used: 1.785 (sec). Leaf size: 120

ode=D[y[x],x]==y[x]^2+\[Lambda]*x*ArcCos[x]^n*y[x]+\[Lambda]*ArcCos[x]^n; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\exp \left (-\int _1^x-\lambda \arccos (K[1])^n K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-\lambda \arccos (K[1])^n K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\lambda \arccos (K[1])^n K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )}\\ y(x)&\to -\frac {1}{x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-lambda_*x*y(x)*acos(x)**n - lambda_*acos(x)**n - y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -lambda_*x*y(x)*acos(x)**n - lambda_*acos(x)**n - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method