88.6.8 problem 8
Internal
problem
ID
[23990]
Book
:
Elementary
Differential
Equations.
By
Lee
Roy
Wilcox
and
Herbert
J.
Curtis.
1961
first
edition.
International
texbook
company.
Scranton,
Penn.
USA.
CAT
number
61-15976
Section
:
Chapter
2.
Differential
equations
of
first
order.
Exercise
at
page
38
Problem
number
:
8
Date
solved
:
Thursday, October 02, 2025 at 09:49:34 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y^{\prime }&=\frac {x^{3}+x^{2} y-y^{3}}{x^{3}-x y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 276
ode:=diff(y(x),x) = (x^3+x^2*y(x)-y(x)^3)/(x^3-x*y(x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (\left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{2}/{3}}+4\right ) x}{2 \left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (i \left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{2}/{3}} \sqrt {3}+\left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}+4\right ) x}{4 \left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {\left (\left (i \sqrt {3}-1\right ) \left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{2}/{3}}-4 i \sqrt {3}-4\right ) x}{4 \left (-12 \ln \left (x \right )-12 c_1 +4 \sqrt {-4+9 \ln \left (x \right )^{2}+18 \ln \left (x \right ) c_1 +9 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 40.872 (sec). Leaf size: 387
ode=D[y[x],{x,1}]==( x^3+x^2*y[x]-y[x]^3 )/( x^3-x*y[x]^2 );
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{\sqrt {x^6 \left (9 \log ^2(x)-18 c_1 \log (x)-4+9 c_1{}^2\right )}-3 x^3 \log (x)+3 c_1 x^3}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {x^6 \left (9 \log ^2(x)-18 c_1 \log (x)-4+9 c_1{}^2\right )}-3 x^3 \log (x)+3 c_1 x^3}}\\ y(x)&\to \frac {i \left (\left (\sqrt {3}+i\right ) \left (2 \sqrt {x^6 \left (9 \log ^2(x)-18 c_1 \log (x)-4+9 c_1{}^2\right )}-6 x^3 \log (x)+6 c_1 x^3\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) x^2\right )}{4 \sqrt [3]{\sqrt {x^6 \left (9 \log ^2(x)-18 c_1 \log (x)-4+9 c_1{}^2\right )}-3 x^3 \log (x)+3 c_1 x^3}}\\ y(x)&\to -\frac {i \left (\left (\sqrt {3}-i\right ) \left (2 \sqrt {x^6 \left (9 \log ^2(x)-18 c_1 \log (x)-4+9 c_1{}^2\right )}-6 x^3 \log (x)+6 c_1 x^3\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2\right )}{4 \sqrt [3]{\sqrt {x^6 \left (9 \log ^2(x)-18 c_1 \log (x)-4+9 c_1{}^2\right )}-3 x^3 \log (x)+3 c_1 x^3}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (x**3 + x**2*y(x) - y(x)**3)/(x**3 - x*y(x)**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
TypeError : cannot determine truth value of Relational: -1 < 3*x**3