23.1.312 problem 300

Internal problem ID [4919]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 300
Date solved : Tuesday, September 30, 2025 at 08:56:38 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

\[ y = x +\frac {1}{c_1 -\operatorname {arctanh}\left (x \right )} \]

✓ Mathematica. Time used: 0.154 (sec). Leaf size: 36

ode=(1-x^2)*D[y[x],x]==1-(2*x-y[x])*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.282 (sec). Leaf size: 134

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

\[ y{\left (x \right )} = \frac {- 3 x^{5} \log {\left (x - 1 \right )} + 3 x^{5} \log {\left (x + 1 \right )} + 10 x^{4} + 6 x^{3} \log {\left (x - 1 \right )} - 6 x^{3} \log {\left (x + 1 \right )} - 22 x^{2} - 3 x \log {\left (x - 1 \right )} + 3 x \log {\left (x + 1 \right )} + 16}{- 3 x^{4} \log {\left (x - 1 \right )} + 3 x^{4} \log {\left (x + 1 \right )} - 6 x^{3} + 6 x^{2} \log {\left (x - 1 \right )} - 6 x^{2} \log {\left (x + 1 \right )} + 10 x - 3 \log {\left (x - 1 \right )} + 3 \log {\left (x + 1 \right )}} \]