54.1.319 problem 325
Internal
problem
ID
[11633]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
325
Date
solved
:
Tuesday, September 30, 2025 at 09:49:22 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y \left (y^{3}-2 x^{3}\right ) y^{\prime }+\left (2 y^{3}-x^{3}\right ) x&=0 \end{align*}
✓ Maple. Time used: 0.415 (sec). Leaf size: 122
ode:=y(x)*(y(x)^3-2*x^3)*diff(y(x),x)+(2*y(x)^3-x^3)*x = 0;
dsolve(ode,y(x), singsol=all);
\[
-\frac {4 \ln \left (2\right )}{7}-\frac {2 \ln \left (\frac {x^{4}+x^{3} y+3 x^{2} y^{2}+y^{3} x +y^{4}}{x^{4}}\right )}{7}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 y+x \right ) \sqrt {3}}{3 x}\right )}{7}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}+4 x^{2} y+2 y^{2} x +2 y^{3}\right )}{3 x^{3}}\right )}{7}+\frac {\ln \left (\frac {y-x}{x}\right )}{7}-\ln \left (x \right )-c_1 = 0
\]
✓ Mathematica. Time used: 0.106 (sec). Leaf size: 139
ode=x*(-x^3 + 2*y[x]^3) + y[x]*(-2*x^3 + y[x]^3)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {1}{7} \text {RootSum}\left [\text {$\#$1}^4+\text {$\#$1}^3+3 \text {$\#$1}^2+\text {$\#$1}+1\&,\frac {8 \text {$\#$1}^3 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+9 \text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+12 \text {$\#$1} \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{4 \text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+1}\&\right ]-\frac {1}{7} \log \left (1-\frac {y(x)}{x}\right )=-\log (x)+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 2.578 (sec). Leaf size: 116
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(-x**3 + 2*y(x)**3) + (-2*x**3 + y(x)**3)*y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\log {\left (x \right )} = C_{1} - \frac {2 \sqrt {3} \left (\operatorname {atan}{\left (\frac {\sqrt {3} \left (1 + \frac {2 y{\left (x \right )}}{x}\right )}{3} \right )} - \operatorname {atan}{\left (\frac {\sqrt {3} \left (1 + \frac {4 y{\left (x \right )}}{x} + \frac {2 y^{2}{\left (x \right )}}{x^{2}} + \frac {2 y^{3}{\left (x \right )}}{x^{3}}\right )}{3} \right )}\right )}{7} + \log {\left (\frac {\sqrt [7]{-1 + \frac {y{\left (x \right )}}{x}}}{\left (1 + \frac {y{\left (x \right )}}{x} + \frac {3 y^{2}{\left (x \right )}}{x^{2}} + \frac {y^{3}{\left (x \right )}}{x^{3}} + \frac {y^{4}{\left (x \right )}}{x^{4}}\right )^{\frac {2}{7}}} \right )}
\]