69.7.2 problem 176

Internal problem ID [18082]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 176
Date solved : Thursday, October 02, 2025 at 02:38:01 PM
CAS classification : [_exact, _rational]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 125

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

\begin{align*} y &= -\frac {\sqrt {-6 x^{2}-2 \sqrt {9 x^{4}-4 x^{3}-4 c_1}}}{2} \\ y &= \frac {\sqrt {-6 x^{2}-2 \sqrt {9 x^{4}-4 x^{3}-4 c_1}}}{2} \\ y &= -\frac {\sqrt {-6 x^{2}+2 \sqrt {9 x^{4}-4 x^{3}-4 c_1}}}{2} \\ y &= \frac {\sqrt {-6 x^{2}+2 \sqrt {9 x^{4}-4 x^{3}-4 c_1}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 4.605 (sec). Leaf size: 163

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

\begin{align*} y(x)&\to -\frac {\sqrt {-3 x^2-\sqrt {9 x^4-4 x^3+4 c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-3 x^2-\sqrt {9 x^4-4 x^3+4 c_1}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-3 x^2+\sqrt {9 x^4-4 x^3+4 c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-3 x^2+\sqrt {9 x^4-4 x^3+4 c_1}}}{\sqrt {2}} \end{align*}

✗ Sympy

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

Timed Out