89.3.14 problem 14
Internal
problem
ID
[24312]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
2.
Equations
of
the
first
order
and
first
degree.
Exercises
at
page
34
Problem
number
:
14
Date
solved
:
Thursday, October 02, 2025 at 10:16:50 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (6+3 y x -4 y^{3}\right ) x +\left (x^{3}-6 x^{2} y^{2}-1\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 490
ode:=x*(3*x*y(x)-4*y(x)^3+6)+(x^3-6*x^2*y(x)^2-1)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {6 x^{3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{2}/{3}}-6}{6 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{1}/{3}}} \\
y &= \frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{2}/{3}} \sqrt {3}-6 x^{3}-\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{2}/{3}}-6 i \sqrt {3}+6}{12 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{1}/{3}}} \\
y &= -\frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{2}/{3}} \sqrt {3}+6 x^{3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{2}/{3}}-6 i \sqrt {3}-6}{12 \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}+162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}+54 c_1 x \right )^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 60.072 (sec). Leaf size: 424
ode=x*( 3*x*y[x]-4*y[x]^3+6 )+ ( x^3-6*x^2*y[x]^2-1 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {\sqrt [3]{2} \left (x^3-1\right )}{\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}-\frac {\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{6 \sqrt [3]{2} x^2}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(3*x*y(x) - 4*y(x)**3 + 6) + (x**3 - 6*x**2*y(x)**2 - 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out