55.15.3 problem 12

Internal problem ID [13465]
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 : 12
Date solved : Wednesday, October 01, 2025 at 02:20:11 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-\left (1+k \right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{1+k} y-1\right ) \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 155

ode:=diff(y(x),x) = -(1+k)*x^k*y(x)^2+lambda*arccos(x)^n*(x^(1+k)*y(x)-1); 
dsolve(ode,y(x), singsol=all);
 

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

✗ Mathematica

ode=D[y[x],x]==-(k+1)*x^k*y[x]^2+\[Lambda]*ArcCos[x]^n*(x^(k+1)*y[x]-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
k = symbols("k") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-lambda_*(x**(k + 1)*y(x) - 1)*acos(x)**n + x**k*(k + 1)*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out