23.2.14 problem 14

Internal problem ID [5369]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 12:36:18 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 22

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

\begin{align*} y &= 1 \\ y &= 0 \\ y &= \sec \left (-\frac {x}{2}+\frac {c_1}{2}\right )^{2} \\ \end{align*}

✓ Mathematica. Time used: 0.672 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to \sec ^2\left (\frac {x-c_1}{2}\right )\\ y(x)&\to 1+\tan ^2\left (\frac {x+c_1}{2}\right )\\ y(x)&\to 0\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.457 (sec). Leaf size: 68

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

\[ \left [ \begin {cases} 2 i \operatorname {acosh}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {for}\: \frac {1}{\left |{y{\left (x \right )}}\right |} > 1 \\- 2 \operatorname {asin}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {otherwise} \end {cases} = C_{1} - x, \ \begin {cases} 2 i \operatorname {acosh}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {for}\: \frac {1}{\left |{y{\left (x \right )}}\right |} > 1 \\- 2 \operatorname {asin}{\left (\frac {1}{\sqrt {y{\left (x \right )}}} \right )} & \text {otherwise} \end {cases} = C_{1} + x\right ] \]