23.1.350 problem 336

Internal problem ID [4957]
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 : 336
Date solved : Tuesday, September 30, 2025 at 09:05:40 AM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} x \left (1-2 x \right ) y^{\prime }&=4 x -\left (1+4 x \right ) y+y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=x*(1-2*x)*diff(y(x),x) = 4*x-(1+4*x)*y(x)+y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-2 x^{2}+c_1}{c_1 -x} \]

✓ Mathematica. Time used: 0.25 (sec). Leaf size: 94

ode=x*(1-2*x)*D[y[x],x]==4*x -(1+4*x)*y[x]+y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 1+\frac {\exp \left (\int _1^x\frac {1-4 K[1]}{K[1]-2 K[1]^2}dK[1]\right )}{-\int _1^x\frac {\exp \left (\int _1^{K[2]}\frac {1-4 K[1]}{K[1]-2 K[1]^2}dK[1]\right )}{K[2]-2 K[2]^2}dK[2]+c_1}\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.280 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - 2*x)*Derivative(y(x), x) - 4*x + (4*x + 1)*y(x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} + 2 x^{2}}{C_{1} + x} \]