89.6.4 problem 4

Internal problem ID [24387]
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 : 4
Date solved : Thursday, October 02, 2025 at 10:23:33 PM
CAS classification : [_separable]

\begin{align*} x^{2}+1+x^{2} y^{2} y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 88

ode:=x^2+1+x^2*y(x)^2*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.251 (sec). Leaf size: 89

ode=(x^2+1)+x^2*y[x]^2*D[y[x],x]== 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.115 (sec). Leaf size: 66

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

\[ \left [ y{\left (x \right )} = \frac {\left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - x + \frac {1}{x}}}{2}, \ y{\left (x \right )} = \frac {\left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - x + \frac {1}{x}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1} - 3 x + \frac {3}{x}}\right ] \]