89.6.44 problem 45

Internal problem ID [24427]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 45
Date solved : Thursday, October 02, 2025 at 10:27:48 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 119

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

\begin{align*} y &= \frac {\sqrt {3 c_1 \,x^{2}-\sqrt {8 x^{4} c_1^{2}+1}}}{\sqrt {c_1}} \\ y &= \frac {\sqrt {3 c_1 \,x^{2}+\sqrt {8 x^{4} c_1^{2}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {3 c_1 \,x^{2}-\sqrt {8 x^{4} c_1^{2}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {3 c_1 \,x^{2}+\sqrt {8 x^{4} c_1^{2}+1}}}{\sqrt {c_1}} \\ \end{align*}

✓ Mathematica. Time used: 8.416 (sec). Leaf size: 245

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

\begin{align*} y(x)&\to -\sqrt {3 x^2-\sqrt {8 x^4+e^{4 c_1}}}\\ y(x)&\to \sqrt {3 x^2-\sqrt {8 x^4+e^{4 c_1}}}\\ y(x)&\to -\sqrt {3 x^2+\sqrt {8 x^4+e^{4 c_1}}}\\ y(x)&\to \sqrt {3 x^2+\sqrt {8 x^4+e^{4 c_1}}}\\ y(x)&\to -\sqrt {3 x^2-2 \sqrt {2} \sqrt {x^4}}\\ y(x)&\to \sqrt {3 x^2-2 \sqrt {2} \sqrt {x^4}}\\ y(x)&\to -\sqrt {2 \sqrt {2} \sqrt {x^4}+3 x^2}\\ y(x)&\to \sqrt {2 \sqrt {2} \sqrt {x^4}+3 x^2} \end{align*}

✓ Sympy. Time used: 2.568 (sec). Leaf size: 85

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

\[ \left [ y{\left (x \right )} = - \sqrt {3 x^{2} - \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = \sqrt {3 x^{2} - \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = - \sqrt {3 x^{2} + \sqrt {C_{1} + 8 x^{4}}}, \ y{\left (x \right )} = \sqrt {3 x^{2} + \sqrt {C_{1} + 8 x^{4}}}\right ] \]