58.7.4 problem 4
Internal
problem
ID
[14659]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
2,
Section
2.4.
Special
integrating
factors
and
transformations.
Exercises
page
67
Problem
number
:
4
Date
solved
:
Thursday, October 02, 2025 at 09:48:56 AM
CAS
classification
:
[_rational]
\begin{align*} 2 x y^{2}+y+\left (2 y^{3}-x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 297
ode:=2*x*y(x)^2+y(x)+(2*y(x)^3-x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {-12 x^{2}-12 c_1 +\left (-108 x +12 \sqrt {12 x^{6}+36 c_1 \,x^{4}+\left (36 c_1^{2}+81\right ) x^{2}+12 c_1^{3}}\right )^{{2}/{3}}}{6 \left (-108 x +12 \sqrt {12 x^{6}+36 c_1 \,x^{4}+\left (36 c_1^{2}+81\right ) x^{2}+12 c_1^{3}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108 x +12 \sqrt {12 x^{6}+36 c_1 \,x^{4}+\left (36 c_1^{2}+81\right ) x^{2}+12 c_1^{3}}\right )^{{2}/{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+c_1 \right )}{\left (-108 x +12 \sqrt {12 x^{6}+36 c_1 \,x^{4}+\left (36 c_1^{2}+81\right ) x^{2}+12 c_1^{3}}\right )^{{1}/{3}}} \\
y &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-108 x +12 \sqrt {12 x^{6}+36 c_1 \,x^{4}+\left (36 c_1^{2}+81\right ) x^{2}+12 c_1^{3}}\right )^{{2}/{3}}}{12}+\left (1+i \sqrt {3}\right ) \left (x^{2}+c_1 \right )}{\left (-108 x +12 \sqrt {12 x^{6}+36 c_1 \,x^{4}+\left (36 c_1^{2}+81\right ) x^{2}+12 c_1^{3}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 5.615 (sec). Leaf size: 316
ode=(2*x*y[x]^2+y[x])+(2*y[x]^3-x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2^{2/3} \left (-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}\right ){}^{2/3}-6 \sqrt [3]{2} \left (x^2-c_1\right )}{6 \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \left (x^2-c_1\right )}{2^{2/3} \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}{6 \sqrt [3]{2}}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \left (x^2-c_1\right )}{2^{2/3} \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-27 x+\sqrt {729 x^2+108 \left (x^2-c_1\right ){}^3}}}{6 \sqrt [3]{2}}\\ y(x)&\to 0 \end{align*}
✓ Sympy. Time used: 1.963 (sec). Leaf size: 82
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x)**2 + (-x + 2*y(x)**3)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\sqrt [3]{2} \left (- \frac {2 \left (\frac {3 C_{1}}{2} - 3 x^{2}\right )}{\sqrt [3]{27 x + \sqrt {729 x^{2} - 4 \left (\frac {3 C_{1}}{2} - 3 x^{2}\right )^{3}}}} - \sqrt [3]{2} \sqrt [3]{27 x + \sqrt {729 x^{2} - 4 \left (\frac {3 C_{1}}{2} - 3 x^{2}\right )^{3}}}\right )}{6}
\]