23.2.241 problem 247

Internal problem ID [5596]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 247
Date solved : Tuesday, September 30, 2025 at 01:10:48 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} \left (x^{2}-4 y^{2}\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-4 x^{2}+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.134 (sec). Leaf size: 67

ode:=(x^2-4*y(x)^2)*diff(y(x),x)^2+6*x*y(x)*diff(y(x),x)-4*x^2+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x c_1 \operatorname {RootOf}\left (-2 x c_1 \,\textit {\_Z}^{3}+\textit {\_Z}^{4}-1\right )^{3}+1}{\operatorname {RootOf}\left (-2 x c_1 \,\textit {\_Z}^{3}+\textit {\_Z}^{4}-1\right )^{3} c_1} \\ y &= \frac {c_1 x +\operatorname {RootOf}\left (2 x c_1 \,\textit {\_Z}^{3}+\textit {\_Z}^{4}-1\right )}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 97

ode=(x^2-4 y[x]^2) (D[y[x],x])^2 +6 x y[x] D[y[x],x]-4 x^2+y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {2 K[1]-1}{(K[1]-1) (K[1]+1)}dK[1]=-2 \log (x)+c_1,y(x)\right ]\\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {2 K[2]+1}{(K[2]-1) (K[2]+1)}dK[2]=-2 \log (x)+c_1,y(x)\right ]\\ y(x)&\to -x\\ y(x)&\to x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2 + 6*x*y(x)*Derivative(y(x), x) + (x**2 - 4*y(x)**2)*Derivative(y(x), x)**2 + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out