70.5.33 problem 34
Internal
problem
ID
[18740]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.7
(Substitution
Methods).
Problems
at
page
108
Problem
number
:
34
Date
solved
:
Thursday, October 02, 2025 at 03:30:14 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} x^{\prime }&=\frac {2 x y +x^{2}}{3 y^{2}+2 x y} \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 287
ode:=diff(x(y),y) = (2*x(y)*y+x(y)^2)/(3*y^2+2*x(y)*y);
dsolve(ode,x(y), singsol=all);
\begin{align*}
x &= \frac {12^{{1}/{3}} \left (y 12^{{1}/{3}} c_1 +{\left (y \left (\sqrt {3}\, \sqrt {\frac {27 y^{2} c_1 -4 y}{c_1}}+9 y \right ) c_1^{2}\right )}^{{2}/{3}}\right )}{6 c_1 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 y^{2} c_1 -4 y}{c_1}}+9 y \right ) c_1^{2}\right )}^{{1}/{3}}} \\
x &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 y^{2} c_1 -4 y}{c_1}}+9 y \right ) c_1^{2}\right )}^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) y 2^{{2}/{3}} c_1 \right )}{12 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 y^{2} c_1 -4 y}{c_1}}+9 y \right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
x &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 y^{2} c_1 -4 y}{c_1}}+9 y \right ) c_1^{2}\right )}^{{2}/{3}}+y \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}} c_1 \right )}{12 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 y^{2} c_1 -4 y}{c_1}}+9 y \right ) c_1^{2}\right )}^{{1}/{3}} c_1} \\
\end{align*}
✓ Mathematica. Time used: 43.387 (sec). Leaf size: 420
ode=D[x[y],y]==(2*x[y]*y+x[y]^2)/(3*y^2+2*x[y]*y);
ic={};
DSolve[{ode,ic},x[y],y,IncludeSingularSolutions->True]
\begin{align*} x(y)&\to \frac {\sqrt [3]{2} \left (\sqrt {3} \sqrt {-e^{2 c_1} y^3 \left (-27 y+4 e^{c_1}\right )}+9 e^{c_1} y^2\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} y}{6^{2/3} \sqrt [3]{\sqrt {3} \sqrt {-e^{2 c_1} y^3 \left (-27 y+4 e^{c_1}\right )}+9 e^{c_1} y^2}}\\ x(y)&\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {-e^{2 c_1} y^3 \left (-27 y+4 e^{c_1}\right )}+9 e^{c_1} y^2\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} y}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {-e^{2 c_1} y^3 \left (-27 y+4 e^{c_1}\right )}+9 e^{c_1} y^2}}\\ x(y)&\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {3} \sqrt {-e^{2 c_1} y^3 \left (-27 y+4 e^{c_1}\right )}+9 e^{c_1} y^2\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} y}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {-e^{2 c_1} y^3 \left (-27 y+4 e^{c_1}\right )}+9 e^{c_1} y^2}}\\ x(y)&\to 0 \end{align*}
✗ Sympy
from sympy import *
y = symbols("y")
x = Function("x")
ode = Eq(Derivative(x(y), y) - (2*y*x(y) + x(y)**2)/(3*y**2 + 2*y*x(y)),0)
ics = {}
dsolve(ode,func=x(y),ics=ics)
Timed Out