89.10.8 problem 8
Internal
problem
ID
[24501]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
4.
Additional
topics
on
equations
of
first
order
and
first
degree.
Exercises
at
page
77
Problem
number
:
8
Date
solved
:
Thursday, October 02, 2025 at 10:43:18 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} x^{3} y+\left (3 x^{4}-y^{3}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.148 (sec). Leaf size: 37
ode:=x^3*y(x)+(3*x^4-y(x)^3)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
\ln \left (x \right )-c_1 +\frac {\ln \left (\frac {-15 x^{4}+4 y^{3}}{x^{4}}\right )}{20}+\frac {3 \ln \left (\frac {y}{x^{{4}/{3}}}\right )}{5} = 0
\]
✓ Mathematica. Time used: 97.2 (sec). Leaf size: 531
ode=( x^3*y[x])+( 3*x^4-y[x]^3 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,1\right ]}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,1\right ]}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,1\right ]}\\ y(x)&\to \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,2\right ]}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,2\right ]}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,2\right ]}\\ y(x)&\to \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,3\right ]}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,3\right ]}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,3\right ]}\\ y(x)&\to \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,4\right ]}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,4\right ]}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,4\right ]}\\ y(x)&\to \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,5\right ]}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,5\right ]}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{\text {Root}\left [4 \text {$\#$1}^5-15 \text {$\#$1}^4 x^4+60 c_1\&,5\right ]} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3*y(x) + (3*x**4 - y(x)**3)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE x**3*y(x)/(3*x**4 - y(x)**3) + Derivative(y(x), x) cannot be sol