56.10.1 problem Ex 1

Internal problem ID [14119]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 17. Other forms which Integrating factors can be found. Page 25
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:14:55 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 63

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

\begin{align*} y &= \frac {-4 c_1 \,x^{2}-\sqrt {-2 x^{4} c_1^{2}+6}}{6 c_1 x} \\ y &= \frac {-4 c_1 \,x^{2}+\sqrt {-2 x^{4} c_1^{2}+6}}{6 c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 1.023 (sec). Leaf size: 135

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

\begin{align*} y(x)&\to -\frac {4 x^2+\sqrt {-2 x^4+6 e^{4 c_1}}}{6 x}\\ y(x)&\to \frac {-4 x^2+\sqrt {-2 x^4+6 e^{4 c_1}}}{6 x}\\ y(x)&\to -\frac {\sqrt {2} \sqrt {-x^4}+4 x^2}{6 x}\\ y(x)&\to \frac {\sqrt {2} \sqrt {-x^4}-4 x^2}{6 x} \end{align*}

✓ Sympy. Time used: 1.079 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = - \frac {2 x}{3} - \frac {\sqrt {C_{1} - 2 x^{4}}}{6 x}, \ y{\left (x \right )} = - \frac {2 x}{3} + \frac {\sqrt {C_{1} - 2 x^{4}}}{6 x}\right ] \]