19.3.8 problem 16

Internal problem ID [3551]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.8, page 68
Problem number : 16
Date solved : Tuesday, March 04, 2025 at 04:47:12 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.890 (sec). Leaf size: 80

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

\[ y \left (x \right ) = \frac {x \left (-\operatorname {RootOf}\left (2 \textit {\_Z}^{6}+\left (9 c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-6 c_{1} \textit {\_Z}^{2} x^{2}+c_{1} x^{2}\right )^{2}+1\right )}{\operatorname {RootOf}\left (2 \textit {\_Z}^{6}+\left (9 c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-6 c_{1} \textit {\_Z}^{2} x^{2}+c_{1} x^{2}\right )^{2}} \]

✓ Mathematica. Time used: 60.197 (sec). Leaf size: 373

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

\begin{align*} y(x)\to \frac {\sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{3 \sqrt [3]{2}}-\frac {\sqrt [3]{2} \left (-3 x^2+e^{2 c_1}\right )}{\sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{6 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (-3 x^2+e^{2 c_1}\right )}{2^{2/3} \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}{6 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (-3 x^2+e^{2 c_1}\right )}{2^{2/3} \sqrt [3]{-54 x^3+2 \sqrt {729 x^6+\left (-9 x^2+3 e^{2 c_1}\right ){}^3}}}+x \\ \end{align*}

✗ Sympy

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

Timed Out