30.4.13 problem 13
Internal
problem
ID
[7496]
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
:
13
Date
solved
:
Tuesday, September 30, 2025 at 04:39:42 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} 2 y^{2}-6 x y+\left (3 x y-4 x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 6.311 (sec). Leaf size: 28
ode:=2*y(x)^2-6*x*y(x)+(3*x*y(x)-4*x^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \operatorname {RootOf}\left (x^{5} c_1 \,\textit {\_Z}^{15}-2 x^{5} c_1 \,\textit {\_Z}^{10}-1\right )^{5} x
\]
✓ Mathematica. Time used: 60.092 (sec). Leaf size: 419
ode=( 2*y[x]^2-6*x*y[x] )+( 3*x*y[x]-4*x^2 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{6} \left (\frac {8 \sqrt [3]{2} x^2}{\sqrt [3]{\frac {16 x^5+3 \sqrt {3} \sqrt {e^{c_1} \left (-32 x^5+27 e^{c_1}\right )}-27 e^{c_1}}{x^2}}}+2^{2/3} \sqrt [3]{\frac {16 x^5+3 \sqrt {3} \sqrt {e^{c_1} \left (-32 x^5+27 e^{c_1}\right )}-27 e^{c_1}}{x^2}}+4 x\right )\\ y(x)&\to \frac {1}{12} \left (-\frac {8 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{\frac {16 x^5+3 \sqrt {3} \sqrt {e^{c_1} \left (-32 x^5+27 e^{c_1}\right )}-27 e^{c_1}}{x^2}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\frac {16 x^5+3 \sqrt {3} \sqrt {e^{c_1} \left (-32 x^5+27 e^{c_1}\right )}-27 e^{c_1}}{x^2}}+8 x\right )\\ y(x)&\to \frac {1}{12} \left (\frac {8 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{\frac {16 x^5+3 \sqrt {3} \sqrt {e^{c_1} \left (-32 x^5+27 e^{c_1}\right )}-27 e^{c_1}}{x^2}}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\frac {16 x^5+3 \sqrt {3} \sqrt {e^{c_1} \left (-32 x^5+27 e^{c_1}\right )}-27 e^{c_1}}{x^2}}+8 x\right ) \end{align*}
✓ Sympy. Time used: 49.873 (sec). Leaf size: 389
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-6*x*y(x) + (-4*x**2 + 3*x*y(x))*Derivative(y(x), x) + 2*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {8 x^{2}}{\sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}}} + 2 x - 2 \sqrt {3} i x - \sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}} - \sqrt {3} i \sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}}}{3 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {\frac {8 x^{2}}{\sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}}} + 2 x + 2 \sqrt {3} i x - \sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}} + \sqrt {3} i \sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}}}{3 \left (1 + \sqrt {3} i\right )}, \ y{\left (x \right )} = - \frac {4 x^{2}}{3 \sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}}} + \frac {2 x}{3} - \frac {\sqrt [3]{\frac {27 C_{1}}{x^{2}} - 8 x^{3} + 3 \sqrt {3} \sqrt {C_{1} \left (\frac {27 C_{1}}{x^{4}} - 16 x\right )}}}{3}\right ]
\]