85.33.1 problem 1

Internal problem ID [22624]
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 65
Problem number : 1
Date solved : Thursday, October 02, 2025 at 08:56:48 PM
CAS classification : [_separable]

\begin{align*} \left (x^{2}+1\right ) \left (y^{3}-1\right )&=x^{2} y^{2} y^{\prime } \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 154

ode:=(x^2+1)*(y(x)^3-1) = x^2*y(x)^2*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left ({\mathrm e}^{-\frac {9 \left (x -1\right ) \left (x +1\right )}{x}}+{\mathrm e}^{-\frac {6 \left (x -1\right ) \left (x +1\right )}{x}} c_1 \right )^{{1}/{3}} {\mathrm e}^{\frac {3 \left (x -1\right ) \left (x +1\right )}{x}} \\ y &= -\frac {\left ({\mathrm e}^{-\frac {9 \left (x -1\right ) \left (x +1\right )}{x}}+{\mathrm e}^{-\frac {6 \left (x -1\right ) \left (x +1\right )}{x}} c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {3 \left (x -1\right ) \left (x +1\right )}{x}}}{2} \\ y &= \frac {\left ({\mathrm e}^{-\frac {9 \left (x -1\right ) \left (x +1\right )}{x}}+{\mathrm e}^{-\frac {6 \left (x -1\right ) \left (x +1\right )}{x}} c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\mathrm e}^{\frac {3 \left (x -1\right ) \left (x +1\right )}{x}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 1.925 (sec). Leaf size: 108

ode=(x^2+1)*(y[x]^3-1)==x^2*y[x]^2*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt [3]{1+e^{3 \left (x-\frac {1}{x}+c_1\right )}}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{1+e^{3 \left (x-\frac {1}{x}+c_1\right )}}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{1+e^{3 \left (x-\frac {1}{x}+c_1\right )}}\\ y(x)&\to 1\\ y(x)&\to -\sqrt [3]{-1}\\ y(x)&\to (-1)^{2/3} \end{align*}

✓ Sympy. Time used: 2.056 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*y(x)**2*Derivative(y(x), x) + (x**2 + 1)*(y(x)**3 - 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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