26.6.43 problem Exercise 12.43, page 103
Internal
problem
ID
[7043]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
12,
Miscellaneous
Methods
Problem
number
:
Exercise
12.43,
page
103
Date
solved
:
Tuesday, September 30, 2025 at 04:17:10 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4}&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 317
ode:=(2*x*y(x)^3-x^4)*diff(y(x),x)+2*x^3*y(x)-y(x)^4 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {12^{{1}/{3}} \left (x 12^{{1}/{3}} c_1 +{\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{2}/{3}}\right )}{6 c_1 {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{1}/{3}}} \\
y &= \frac {\left (\left (-i \sqrt {3}-1\right ) {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{2}/{3}}+c_1 x \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) 2^{{2}/{3}}\right ) 2^{{2}/{3}} 3^{{1}/{3}}}{12 {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
y &= -\frac {\left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{2}/{3}}+c_1 x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}\right ) 2^{{2}/{3}} 3^{{1}/{3}}}{12 {\left (x \left (-9 c_1 \,x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_1^{3} x^{4}-4 x}{c_1}}\right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 60.138 (sec). Leaf size: 331
ode=(2*x*y[x]^3-x^4)*D[y[x],x]+2*x^3*y[x]-y[x]^4==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}}\\ y(x)&\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}}\\ y(x)&\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \end{align*}
✓ Sympy. Time used: 4.401 (sec). Leaf size: 5
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x**3*y(x) + (-x**4 + 2*x*y(x)**3)*Derivative(y(x), x) - y(x)**4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} x
\]