54.1.162 problem 165

Internal problem ID [11476]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 165
Date solved : Tuesday, September 30, 2025 at 08:31:18 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} x \left (2 x -1\right ) y^{\prime }+y^{2}-\left (4 x +1\right ) y+4 x&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 17

ode:=x*(2*x-1)*diff(y(x),x)+y(x)^2-(1+4*x)*y(x)+4*x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.188 (sec). Leaf size: 94

ode=x*(2*x-1)*D[y[x],x] + y[x]^2 - (4*x+1)*y[x] + 4*x==0; 
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: 0.270 (sec). Leaf size: 12

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

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