79.2.2 problem (c)

Internal problem ID [21084]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises XIII at page 24
Problem number : (c)
Date solved : Thursday, October 02, 2025 at 07:06:59 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

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

\[ y = \frac {1}{2}+\frac {\sin \left (x +c_1 \right )}{2} \]

✓ Mathematica. Time used: 0.22 (sec). Leaf size: 34

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

\begin{align*} y(x)&\to \sin ^2\left (\frac {x+c_1}{2}\right )\\ y(x)&\to 0\\ y(x)&\to 1\\ y(x)&\to \text {Interval}[\{0,1\}] \end{align*}

✓ Sympy. Time used: 0.231 (sec). Leaf size: 12

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

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