6.5.55 problem 56

Internal problem ID [1679]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 56
Date solved : Tuesday, September 30, 2025 at 04:59:04 AM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 19

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

\[ y = 1-\frac {2}{{\mathrm e}^{x} c_1 -2 x -2} \]

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to \frac {x+c_1 e^x+2}{x+c_1 e^x+1}\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.210 (sec). Leaf size: 20

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

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