56.15.3 problem Ex 3

Internal problem ID [14171]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 26. Equations solvable for \(x\). Page 55
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:18:03 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} {y^{\prime }}^{2} x -2 y y^{\prime }-x&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 32

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \frac {-c_1^{2}+x^{2}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 45

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

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

✓ Sympy. Time used: 5.735 (sec). Leaf size: 22

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

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