23.1.678 problem 673

Internal problem ID [5285]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 673
Date solved : Tuesday, September 30, 2025 at 12:07:18 PM
CAS classification : [_exact, _rational]

\begin{align*} \left (1-x^{3}+6 x^{2} y^{2}\right ) y^{\prime }&=\left (6+3 x y-4 y^{3}\right ) x \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 484

ode:=(1-x^3+6*x^2*y(x)^2)*diff(y(x),x) = (6+3*x*y(x)-4*y(x)^3)*x; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {6 x^{3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{2}/{3}}-6}{6 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{1}/{3}}} \\ y &= \frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{2}/{3}} \sqrt {3}-6 x^{3}-\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{2}/{3}}-6 i \sqrt {3}+6}{12 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{1}/{3}}} \\ y &= -\frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{2}/{3}} \sqrt {3}+6 x^{3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{2}/{3}}-6 i \sqrt {3}-6}{12 \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_1 \,x^{4}+27 c_1^{2} x^{2}-6 x^{3}+2}-54 c_1 x \right )^{{1}/{3}} x} \\ \end{align*}

✓ Mathematica. Time used: 60.079 (sec). Leaf size: 424

ode=(1-x^3+6 x^2 y[x]^2)D[y[x],x]==(6+3 x y[x]-4 y[x]^3)x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt [3]{2} \left (x^3-1\right )}{\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}-\frac {\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{6 \sqrt [3]{2} x^2}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(3*x*y(x) - 4*y(x)**3 + 6) + (-x**3 + 6*x**2*y(x)**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out