61.14.7 problem 7

Internal problem ID [12161]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-1. Equations containing arcsine.
Problem number : 7
Date solved : Sunday, March 30, 2025 at 11:25:15 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 24

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

\[ y = a \,x^{m}+b +\frac {1}{c_1 -\lambda \int \arcsin \left (x \right )^{n}d x} \]

✓ Mathematica. Time used: 0.87 (sec). Leaf size: 44

ode=D[y[x],x]==\[Lambda]*ArcSin[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 \arcsin (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*asin(x)**n + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out