29.26.25 problem 761

Internal problem ID [5346]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 761
Date solved : Tuesday, March 04, 2025 at 09:29:56 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 1.066 (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: 0.787 (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)
 

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