84.12.5 problem 6.10

Internal problem ID [22147]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 6. Exact first-order differential equations. Supplementary problems
Problem number : 6.10
Date solved : Thursday, October 02, 2025 at 08:32:26 PM
CAS classification : [[_homogeneous, `class G`], _exact, _rational]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 101

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

\begin{align*} y &= 0 \\ y &= -\frac {\sqrt {-2 x^{3}-2 \sqrt {x^{6}-4 c_1}}}{2} \\ y &= \frac {\sqrt {-2 x^{3}-2 \sqrt {x^{6}-4 c_1}}}{2} \\ y &= -\frac {\sqrt {-2 x^{3}+2 \sqrt {x^{6}-4 c_1}}}{2} \\ y &= \frac {\sqrt {-2 x^{3}+2 \sqrt {x^{6}-4 c_1}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 2.062 (sec). Leaf size: 145

ode=3*x^2*y[x]^2+(2*x^3*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 0\\ y(x)&\to -\frac {\sqrt {-x^3-\sqrt {x^6+4 c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^3-\sqrt {x^6+4 c_1}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {-x^3+\sqrt {x^6+4 c_1}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-x^3+\sqrt {x^6+4 c_1}}}{\sqrt {2}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 3.386 (sec). Leaf size: 102

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

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