76.5.33 problem 34
Internal
problem
ID
[17374]
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, March 13, 2025 at 10:05:45 AM
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: 0.075 (sec). Leaf size: 283
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 \left (y \right ) &= \frac {12^{{1}/{3}} \left (y 12^{{1}/{3}} c_{1} +{\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-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 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{1}/{3}}} \\
x \left (y \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-i \sqrt {3}-1\right ) {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} y \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_{1} \right )}{12 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
x \left (y \right ) &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} y \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) c_{1} \right ) 3^{{1}/{3}}}{12 {\left (y \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} y^{2}-4 y}{c_{1}}}+9 y \right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Mathematica. Time used: 0.134 (sec). Leaf size: 40
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]
\[
\text {Solve}\left [\int _1^{\frac {x(y)}{y}}\frac {2 K[1]+3}{K[1] (K[1]+1)}dK[1]=-\log (y)+c_1,x(y)\right ]
\]
✗ 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