60.1.397 problem 408
Internal
problem
ID
[10411]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
408
Date
solved
:
Sunday, March 30, 2025 at 04:40:52 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}-2 y+x&=0 \end{align*}
✓ Maple. Time used: 0.031 (sec). Leaf size: 96
ode:=x*diff(y(x),x)^2-2*y(x)+x = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (2 \operatorname {LambertW}\left (\frac {\sqrt {c_1 x}}{c_1}\right )^{2}+2 \operatorname {LambertW}\left (\frac {\sqrt {c_1 x}}{c_1}\right )+1\right ) x}{2 \operatorname {LambertW}\left (\frac {\sqrt {c_1 x}}{c_1}\right )^{2}} \\
y &= \frac {\left (2 \operatorname {LambertW}\left (-\frac {\sqrt {c_1 x}}{c_1}\right )^{2}+2 \operatorname {LambertW}\left (-\frac {\sqrt {c_1 x}}{c_1}\right )+1\right ) x}{2 \operatorname {LambertW}\left (-\frac {\sqrt {c_1 x}}{c_1}\right )^{2}} \\
\end{align*}
✓ Mathematica. Time used: 0.557 (sec). Leaf size: 97
ode=x - 2*y[x] + x*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}-1}-2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}-1\right )&=\log (x)+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {2}{\sqrt {\frac {2 y(x)}{x}-1}+1}+2 \log \left (\sqrt {\frac {2 y(x)}{x}-1}+1\right )&=-\log (x)+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 6.882 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x)**2 + x - 2*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} - \log {\left (- \sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} + \frac {y{\left (x \right )}}{x} \right )} + \frac {2}{\sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} - 1}, \ \log {\left (x \right )} = C_{1} - \log {\left (\sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} + \frac {y{\left (x \right )}}{x} \right )} - \frac {2}{\sqrt {-1 + \frac {2 y{\left (x \right )}}{x}} + 1}\right ]
\]