85.9.3 problem 1 (c)

Internal problem ID [22477]
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 37
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 08:40:38 PM
CAS classification : [_separable]

\begin{align*} 3 x \left (1+y^{2}\right )+y \left (x^{2}+2\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 76

ode:=3*x*(1+y(x)^2)+y(x)*(x^2+2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\left (x^{2}+2\right ) \left (-x^{6}-6 x^{4}-12 x^{2}+c_1 -8\right )}}{\left (x^{2}+2\right )^{2}} \\ y &= -\frac {\sqrt {\left (x^{2}+2\right ) \left (-x^{6}-6 x^{4}-12 x^{2}+c_1 -8\right )}}{\left (x^{2}+2\right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 8.91 (sec). Leaf size: 138

ode=3*x*(y[x]^2+1)+y[x]*(x^2+2)*D[y[x],{x,1}]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.716 (sec). Leaf size: 49

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {C_{1}}{x^{6} + 6 x^{4} + 12 x^{2} + 8} - 1}, \ y{\left (x \right )} = \sqrt {\frac {C_{1}}{x^{6} + 6 x^{4} + 12 x^{2} + 8} - 1}\right ] \]