54.1.474 problem 487

Internal problem ID [11788]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 487
Date solved : Tuesday, September 30, 2025 at 10:36:42 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 105

ode:=y(x)^2*diff(y(x),x)^2-6*x^3*diff(y(x),x)+4*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {18^{{1}/{3}} x^{{4}/{3}}}{2} \\ y &= -\frac {18^{{1}/{3}} x^{{4}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {18^{{1}/{3}} x^{{4}/{3}} \left (i \sqrt {3}-1\right )}{4} \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-3 \int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} +4 c_1 \right ) x^{{4}/{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.998 (sec). Leaf size: 218

ode=4*x^2*y[x] - 6*x^3*D[y[x],x] + y[x]^2*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {3}{4} \log (y(x))-\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \text {arctanh}\left (\frac {\sqrt {9 x^4-4 y(x)^3}}{3 x^2}\right )}{2 x \sqrt {9 x^4-4 y(x)^3}}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {9 x^6-4 x^2 y(x)^3} \text {arctanh}\left (\frac {\sqrt {9 x^4-4 y(x)^3}}{3 x^2}\right )}{2 x \sqrt {9 x^4-4 y(x)^3}}+\frac {3}{4} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to \left (-\frac {3}{2}\right )^{2/3} x^{4/3}\\ y(x)&\to \left (\frac {3}{2}\right )^{2/3} x^{4/3}\\ y(x)&\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} x^{4/3} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*x**3*Derivative(y(x), x) + 4*x**2*y(x) + y(x)**2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out