61.14.2 problem 2

Internal problem ID [12156]
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 : 2
Date solved : Sunday, March 30, 2025 at 11:22:39 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 55

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

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

✓ Mathematica. Time used: 1.681 (sec). Leaf size: 256

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

\begin{align*} y(x)\to -\frac {\int _1^x\frac {\exp \left (-2^{-n-3} \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left (\Gamma (n+1,2 i \arcsin (K[1])) (-i \arcsin (K[1]))^n+(i \arcsin (K[1]))^n \Gamma (n+1,-2 i \arcsin (K[1]))\right )\right )}{K[1]^2}dK[1]+\frac {\exp \left (\lambda \left (-2^{-n-3}\right ) \arcsin (x)^n \left (\arcsin (x)^2\right )^{-n} \left ((-i \arcsin (x))^n \Gamma (n+1,2 i \arcsin (x))+(i \arcsin (x))^n \Gamma (n+1,-2 i \arcsin (x))\right )\right )}{x}+c_1}{x \left (\int _1^x\frac {\exp \left (-2^{-n-3} \lambda \arcsin (K[1])^n \left (\arcsin (K[1])^2\right )^{-n} \left (\Gamma (n+1,2 i \arcsin (K[1])) (-i \arcsin (K[1]))^n+(i \arcsin (K[1]))^n \Gamma (n+1,-2 i \arcsin (K[1]))\right )\right )}{K[1]^2}dK[1]+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)*asin(x)**n - lambda_*asin(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)*asin(x)**n - lambda_*asin(x)**n - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method