31.6.4 problem 4

Internal problem ID [5753]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 4
Date solved : Sunday, March 30, 2025 at 10:08:08 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} {y^{\prime }}^{2}+\frac {2 x y^{\prime }}{y}-1&=0 \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 43

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= -\frac {2 \sqrt {c_1 x +1}}{c_1} \\ y &= \frac {2 \sqrt {c_1 x +1}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.47 (sec). Leaf size: 126

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

\begin{align*} y(x)\to -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}} \\ y(x)\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}

✗ Sympy

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

Timed Out