60.1.454 problem 467
Internal
problem
ID
[10468]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
467
Date
solved
:
Sunday, March 30, 2025 at 04:50:53 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.155 (sec). Leaf size: 92
ode:=y(x)*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} +c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} +c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 0.615 (sec). Leaf size: 65
ode=y[x] - 4*x*D[y[x],x] + y[x]*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \left (y(x)\text {/.}\, \left \{4 x&=\text {K$\$$15315806} y(x)+\frac {y(x)}{\text {K$\$$15315806}},c_1 \exp \left (\int _1^{\text {K$\$$15315806}}\frac {1-K[1]^2}{K[1] \left (K[1]^2-3\right )}dK[1]\right )=y(x)\right \}\right ) \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 32.429 (sec). Leaf size: 105
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-4*x*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \sqrt {3} \sqrt {x^{2}}, \ y{\left (x \right )} = \sqrt {3} \sqrt {x^{2}}, \ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt {2} \sqrt [6]{\sqrt {4 - \frac {y^{2}{\left (x \right )}}{x^{2}}} - 1}}{2 \sqrt [6]{\sqrt {4 - \frac {y^{2}{\left (x \right )}}{x^{2}}} + 2} \sqrt {- \sqrt {4 - \frac {y^{2}{\left (x \right )}}{x^{2}}} - 2 + \frac {y^{2}{\left (x \right )}}{x^{2}}}} \right )}\right ]
\]