70.5.9 problem 9

Internal problem ID [18716]
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 : 9
Date solved : Thursday, October 02, 2025 at 03:23:08 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{\prime }&=\frac {y^{4}+2 x y^{3}-3 x^{2} y^{2}-2 x^{3} y}{2 x^{2} y^{2}-2 x^{3} y-2 x^{4}} \end{align*}

✓ Maple. Time used: 0.075 (sec). Leaf size: 64

ode:=diff(y(x),x) = (y(x)^4+2*x*y(x)^3-3*x^2*y(x)^2-2*x^3*y(x))/(2*x^2*y(x)^2-2*x^3*y(x)-2*x^4); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left ({\mathrm e}^{\operatorname {RootOf}\left (-\ln \left (\frac {\left ({\mathrm e}^{\textit {\_Z}}-1\right )^{2}}{x \left ({\mathrm e}^{\textit {\_Z}}-2\right )}\right ) {\mathrm e}^{\textit {\_Z}}+c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\frac {\left ({\mathrm e}^{\textit {\_Z}}-1\right )^{2}}{x \left ({\mathrm e}^{\textit {\_Z}}-2\right )}\right )-c_1 -\textit {\_Z} +2\right )}-1\right ) x \]

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 44

ode=D[y[x],x]==(y[x]^4+2*x*y[x]^3 -3*x^2*y[x]^2-2*x^3*y[x] )/( 2*x^2*y[x]^2 -2*x^3*y[x] -2*x^4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [-\frac {1}{2} \log \left (1-\frac {y(x)^2}{x^2}\right )-\frac {x}{y(x)}+\log \left (\frac {y(x)}{x}\right )=\frac {\log (x)}{2}+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.363 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (-2*x**3*y(x) - 3*x**2*y(x)**2 + 2*x*y(x)**3 + y(x)**4)/(-2*x**4 - 2*x**3*y(x) + 2*x**2*y(x)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} - \frac {2 x}{y{\left (x \right )}} - \log {\left (\frac {x \left (\frac {x^{2}}{y^{2}{\left (x \right )}} - 1\right )}{y{\left (x \right )}} \right )} \]