15.20.10 problem 10
Internal
problem
ID
[3318]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
38,
page
173
Problem
number
:
10
Date
solved
:
Sunday, March 30, 2025 at 01:35:28 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y&=y^{\prime } x \left (y^{\prime }+1\right ) \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 65
ode:=y(x) = diff(y(x),x)*x*(diff(y(x),x)+1);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {x \left (1+2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_1}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_1}{x}}}\right )^{2}} \\
y &= \frac {x \left (1+2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_1}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_1}{x}}}\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.523 (sec). Leaf size: 102
ode=y[x]==D[y[x],x]*x*(D[y[x],x]+1);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {1}{\sqrt {\frac {4 y(x)}{x}+1}-1}-\log \left (\sqrt {\frac {4 y(x)}{x}+1}-1\right )&=\frac {\log (x)}{2}+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {1}{\sqrt {\frac {4 y(x)}{x}+1}+1}+\log \left (\sqrt {\frac {4 y(x)}{x}+1}+1\right )&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 23.512 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*(Derivative(y(x), x) + 1)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} - \log {\left (\frac {\sqrt {1 + \frac {4 y{\left (x \right )}}{x}}}{2} + \frac {1}{2} + \frac {y{\left (x \right )}}{x} \right )} - \frac {2}{\sqrt {1 + \frac {4 y{\left (x \right )}}{x}} + 1}, \ \log {\left (x \right )} = C_{1} - \log {\left (- \frac {\sqrt {1 + \frac {4 y{\left (x \right )}}{x}}}{2} + \frac {1}{2} + \frac {y{\left (x \right )}}{x} \right )} + \frac {2}{\sqrt {1 + \frac {4 y{\left (x \right )}}{x}} - 1}\right ]
\]