6.3.19 problem 20

Internal problem ID [1596]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 04:39:03 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=2 y-y^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 14

ode:=diff(y(x),x) = 2*y(x)-y(x)^2; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 21

ode=D[y[x],x]==2*y[x]-y[x]^2; 
ic=y[0]==1; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.240 (sec). Leaf size: 12

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

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