85.24.4 problem 4

Internal problem ID [22581]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 57
Problem number : 4
Date solved : Thursday, October 02, 2025 at 08:52:14 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.382 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to \frac {1}{x}-\sqrt {\frac {1}{x^2}} \sqrt {-x^4+c_1 x^2+1}\\ y(x)&\to \frac {1}{x}+\sqrt {\frac {1}{x^2}} \sqrt {-x^4+c_1 x^2+1} \end{align*}

✓ Sympy. Time used: 1.071 (sec). Leaf size: 37

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

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