89.3.2 problem 2
Internal
problem
ID
[24300]
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
:
2
Date
solved
:
Thursday, October 02, 2025 at 10:12:39 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} 6 x +y^{2}+y \left (2 x -3 y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 425
ode:=6*x+y(x)^2+y(x)*(2*x-3*y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36 x^{5}+12 c_1 \,x^{3}+729 x^{4}+486 c_1 \,x^{2}+81 c_1^{2}}\right )^{{1}/{3}}}{6}+\frac {2 x^{2}}{3 \left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36 x^{5}+12 c_1 \,x^{3}+729 x^{4}+486 c_1 \,x^{2}+81 c_1^{2}}\right )^{{1}/{3}}}+\frac {x}{3} \\
y &= \frac {\left (-i \sqrt {3}-1\right ) \left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36}\, \sqrt {\left (x^{3}+\frac {81}{4} x^{2}+\frac {27}{4} c_1 \right ) \left (x^{2}+\frac {c_1}{3}\right )}\right )^{{1}/{3}}}{12}+\frac {x \left (i x \sqrt {3}-x +\left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36}\, \sqrt {\left (x^{3}+\frac {81}{4} x^{2}+\frac {27}{4} c_1 \right ) \left (x^{2}+\frac {c_1}{3}\right )}\right )^{{1}/{3}}\right )}{3 \left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36}\, \sqrt {\left (x^{3}+\frac {81}{4} x^{2}+\frac {27}{4} c_1 \right ) \left (x^{2}+\frac {c_1}{3}\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36}\, \sqrt {\left (x^{3}+\frac {81}{4} x^{2}+\frac {27}{4} c_1 \right ) \left (x^{2}+\frac {c_1}{3}\right )}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{12}-\frac {\left (i x \sqrt {3}+x -\left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36}\, \sqrt {\left (x^{3}+\frac {81}{4} x^{2}+\frac {27}{4} c_1 \right ) \left (x^{2}+\frac {c_1}{3}\right )}\right )^{{1}/{3}}\right ) x}{3 \left (324 x^{2}+108 c_1 +8 x^{3}+12 \sqrt {36}\, \sqrt {\left (x^{3}+\frac {81}{4} x^{2}+\frac {27}{4} c_1 \right ) \left (x^{2}+\frac {c_1}{3}\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 13.307 (sec). Leaf size: 406
ode=(6*x+y[x]^2)+y[x]*( 2*x-3*y[x] )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{6} \left (-\frac {2 \sqrt [3]{2} x^2}{\sqrt [3]{-2 x^3-81 x^2+\sqrt {-4 x^6+\left (2 x^3+81 x^2-27 c_1\right ){}^2}+27 c_1}}-2^{2/3} \sqrt [3]{-2 x^3-81 x^2+\sqrt {-4 x^6+\left (2 x^3+81 x^2-27 c_1\right ){}^2}+27 c_1}+2 x\right )\\ y(x)&\to \frac {1}{12} \left (\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{-2 x^3-81 x^2+\sqrt {-4 x^6+\left (2 x^3+81 x^2-27 c_1\right ){}^2}+27 c_1}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-2 x^3-81 x^2+\sqrt {-4 x^6+\left (2 x^3+81 x^2-27 c_1\right ){}^2}+27 c_1}+4 x\right )\\ y(x)&\to \frac {1}{12} \left (\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{-2 x^3-81 x^2+\sqrt {-4 x^6+\left (2 x^3+81 x^2-27 c_1\right ){}^2}+27 c_1}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3-81 x^2+\sqrt {-4 x^6+\left (2 x^3+81 x^2-27 c_1\right ){}^2}+27 c_1}+4 x\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(6*x + (2*x - 3*y(x))*y(x)*Derivative(y(x), x) + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out