23.1.638 problem 632

Internal problem ID [5245]
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 : 632
Date solved : Tuesday, September 30, 2025 at 12:00:00 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \left (3 x^{2}+2 x y+4 y^{2}\right ) y^{\prime }+2 x^{2}+6 x y+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 432

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

\begin{align*} y &= \frac {\left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{1}/{3}}-\frac {11 x^{2} c_1^{2}}{\left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{1}/{3}}}-c_1 x}{4 c_1} \\ y &= -\frac {11 i \sqrt {3}\, c_1^{2} x^{2}+i \sqrt {3}\, \left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{2}/{3}}-11 x^{2} c_1^{2}+2 c_1 x \left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{1}/{3}}+\left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{2}/{3}}}{8 \left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{1}/{3}} c_1} \\ y &= \frac {11 i \sqrt {3}\, c_1^{2} x^{2}+i \sqrt {3}\, \left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{2}/{3}}+11 x^{2} c_1^{2}-2 c_1 x \left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{1}/{3}}-\left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{2}/{3}}}{8 \left (x^{3} c_1^{3}+8+2 \sqrt {333 x^{6} c_1^{6}+4 x^{3} c_1^{3}+16}\right )^{{1}/{3}} c_1} \\ \end{align*}

✓ Mathematica. Time used: 53.643 (sec). Leaf size: 612

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

\begin{align*} y(x)&\to \frac {1}{4} \left (\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}-\frac {11 x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-x\right )\\ y(x)&\to \frac {1}{16} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right )\\ y(x)&\to \frac {1}{16} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}+\frac {22 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+2 \sqrt {333 x^6+4 e^{3 c_1} x^3+16 e^{6 c_1}}+8 e^{3 c_1}}}-4 x\right )\\ y(x)&\to \frac {1}{4} \left (\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}-\frac {11 x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-x\right )\\ y(x)&\to \frac {1}{8} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}+\frac {11 \left (1-i \sqrt {3}\right ) x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-2 x\right )\\ y(x)&\to \frac {1}{8} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}+\frac {11 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{6 \sqrt {37} \sqrt {x^6}+x^3}}-2 x\right ) \end{align*}

✗ Sympy

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

Timed Out