32.3.1 problem Exact Differential equations. Exercise 9.4, page 79
Internal
problem
ID
[5799]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
9
Problem
number
:
Exact
Differential
equations.
Exercise
9.4,
page
79
Date
solved
:
Sunday, March 30, 2025 at 10:18:14 AM
CAS
classification
:
[_exact, _rational]
\begin{align*} 3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 471
ode:=3*x^2*y(x)+8*x*y(x)^2+(x^3+8*x^2*y(x)+12*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{1}/{3}}}{6}+\frac {x^{3} \left (-3+4 x \right )}{6 \left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{1}/{3}}}-\frac {x^{2}}{3} \\
y &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{2}/{3}}}{4}+x^{2} \left (-\left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{1}/{3}}+\left (i \sqrt {3}-1\right ) \left (-\frac {3}{4}+x \right ) x \right )}{3 \left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (-\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{2}/{3}}+x^{2} \left (\left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{1}/{3}}+\left (-\frac {3}{4}+x \right ) x \left (1+i \sqrt {3}\right )\right )}{3 \left (9 x^{5}-27 c_1 -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_1 \,x^{6}-54 c_1 \,x^{5}+81 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.066 (sec). Leaf size: 474
ode=(3*x^2*y[x]+8*x*y[x]^2)+(x^3+8*x^2*y[x]+12*y[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 (-2 x^2+\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}+\frac {(4 x-3) x^3}{\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}}\right ) \\
y(x)\to \frac {1}{48} \left (-16 x^2+4 i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}-\frac {4 i \left (\sqrt {3}-i\right ) (4 x-3) x^3}{\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}}\right ) \\
y(x)\to \frac {1}{48} \left (-16 x^2-4 \left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}+\frac {4 i \left (\sqrt {3}+i\right ) (4 x-3) x^3}{\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}}\right ) \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*y(x) + 8*x*y(x)**2 + (x**3 + 8*x**2*y(x) + 12*y(x)**2)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out