25.6.4 problem 4
Internal
problem
ID
[6888]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
4
Date
solved
:
Tuesday, September 30, 2025 at 04:00:05 PM
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: 0.27 (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. Time used: 142.511 (sec). Leaf size: 469
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)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}\right ]
\]