30.4.7 problem 7
Internal
problem
ID
[7490]
Book
:
Fundamentals
of
Differential
Equations.
By
Nagle,
Saff
and
Snider.
9th
edition.
Boston.
Pearson
2018.
Section
:
Chapter
2,
First
order
differential
equations.
Section
2.5,
Special
Integrating
Factors.
Exercises.
page
69
Problem
number
:
7
Date
solved
:
Tuesday, September 30, 2025 at 04:39:30 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} 2 x y+\left (y^{2}-3 x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 317
ode:=2*x*y(x)+(y(x)^2-3*x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}}{3 c_1} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 60.112 (sec). Leaf size: 458
ode=( 2*x*y[x] )+( y[x]^2-3*x^2 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{3} \left (\frac {\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1}}{\sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-e^{c_1}\right )\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3}\\ y(x)&\to -\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1}}{3\ 2^{2/3} \sqrt [3]{27 e^{c_1} x^2+3 \sqrt {81 e^{2 c_1} x^4-12 e^{4 c_1} x^2}-2 e^{3 c_1}}}-\frac {e^{c_1}}{3} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x) + (-3*x**2 + y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out