56.20.1 problem Ex 1

Internal problem ID [14196]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter V, Singular solutions. Article 33. Page 73
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:26:01 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 21

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

\begin{align*} y &= x -\ln \left (x \right )+c_1 \\ y &= -x +\ln \left (x \right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to x-\log (x)+c_1\\ y(x)&\to -x+\log (x)+c_1 \end{align*}

✓ Sympy. Time used: 0.221 (sec). Leaf size: 17

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

\[ \left [ y{\left (x \right )} = C_{1} + x - \log {\left (x \right )}, \ y{\left (x \right )} = C_{1} - x + \log {\left (x \right )}\right ] \]