60.1.400 problem 411
Internal
problem
ID
[10414]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
411
Date
solved
:
Wednesday, March 05, 2025 at 10:47:13 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}+x y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 65
ode:=x*diff(y(x),x)^2+x*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (1+2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right ) x}{4 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\
y &= \frac {\left (1+2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right ) x}{4 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.54 (sec). Leaf size: 102
ode=-y[x] + x*D[y[x],x] + x*D[y[x],x]^2==0;
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: 24.275 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x)**2 + x*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} - \log {\left (\frac {\sqrt {1 + \frac {4 y{\left (x \right )}}{x}}}{2} + \frac {1}{2} + \frac {y{\left (x \right )}}{x} \right )} - \frac {2}{\sqrt {1 + \frac {4 y{\left (x \right )}}{x}} + 1}, \ \log {\left (x \right )} = C_{1} - \log {\left (- \frac {\sqrt {1 + \frac {4 y{\left (x \right )}}{x}}}{2} + \frac {1}{2} + \frac {y{\left (x \right )}}{x} \right )} + \frac {2}{\sqrt {1 + \frac {4 y{\left (x \right )}}{x}} - 1}\right ]
\]