81.5.1 problem 6-1

Internal problem ID [21539]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 6. Method of grouping. Page 96.
Problem number : 6-1
Date solved : Thursday, October 02, 2025 at 07:47:12 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 25

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

\[ y = \operatorname {RootOf}\left (x^{9} \textit {\_Z}^{4}+3-{\mathrm e}^{\frac {9 c_1}{4}} \textit {\_Z} \right ) x^{3} \]

✓ Mathematica. Time used: 60.091 (sec). Leaf size: 1021

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

\begin{align*} y(x)&\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}}\\ y(x)&\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}}\\ y(x)&\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}}\\ y(x)&\to \frac {1}{2} \sqrt {-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}}}}-\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {-256 x^9+81 c_1{}^4}}}} \end{align*}

✓ Sympy. Time used: 1.219 (sec). Leaf size: 1034

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

\[ \text {Solution too large to show} \]