76.5.33 problem 34

Internal problem ID [17382]
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 : Monday, March 31, 2025 at 04:10:40 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: 0.008 (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 &= \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 {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}}+y 2^{{2}/{3}} c_1 \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right )\right ) 2^{{2}/{3}}}{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 {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 ) 2^{{2}/{3}}}{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: 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