82.18.9 problem Ex. 10

Internal problem ID [18760]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 10
Date solved : Monday, March 31, 2025 at 06:09:03 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} 3 {y^{\prime }}^{2} y^{2}-2 x y y^{\prime }+4 y^{2}-x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 105

ode:=3*y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+4*y(x)^2-x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {3}\, x}{3} \\ y &= \frac {\sqrt {3}\, x}{3} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y^{2}}{x^{2}}\right )}{2}-c_1 &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y^{2}}{x^{2}}\right )}{2}-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.763 (sec). Leaf size: 179

ode=3*D[y[x],x]^2*y[x]^2-2*x*y[x]*D[y[x],x]+4*y[x]^2-x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to \frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to -\frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to \frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}

✗ Sympy

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

Timed Out