61.15.7 problem 16

Internal problem ID [12170]
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 : 16
Date solved : Monday, March 31, 2025 at 04:02:16 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 165

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

\[ y = \frac {\left (\left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \left (a \,x^{m}+b \right ) \lambda \sqrt {-x^{2}+1}-\left (2+n \right ) \left (\lambda \arccos \left (x \right ) \left (a \,x^{m +1}+b x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\sqrt {\arccos \left (x \right )}\, \left (x^{m} c_1 a +c_1 b +1\right )\right )}{\left (\left (2+n \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right )-\operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (x \right )\right ) \arccos \left (x \right )+\arccos \left (x \right )^{n +\frac {3}{2}}\right ) \lambda \sqrt {-x^{2}+1}+\left (2+n \right ) \left (-\arccos \left (x \right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (x \right )\right ) x \lambda +c_1 \sqrt {\arccos \left (x \right )}\right )} \]

✓ Mathematica. Time used: 0.889 (sec). Leaf size: 44

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

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

✗ Sympy

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

Timed Out