89.6.38 problem 39

Internal problem ID [24421]
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 : 39
Date solved : Thursday, October 02, 2025 at 10:26:57 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.133 (sec). Leaf size: 51

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

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

✓ Sympy. Time used: 0.393 (sec). Leaf size: 42

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

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