89.1.12 problem 12

Internal problem ID [24247]
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. Exercises at page 21
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:01:35 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 32

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}}+4 \,{\mathrm e}^{\textit {\_Z}}+\ln \left (x^{2}+1\right )+2 c_1 -4 \textit {\_Z} -3\right )}-1 \]

✓ Mathematica. Time used: 0.269 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1}^2-2 \text {$\#$1}+4 \log (\text {$\#$1}+1)-3\right )\&\right ]\left [\frac {1}{2} \log \left (x^2+1\right )+c_1\right ]\\ y(x)&\to -1 \end{align*}

✓ Sympy. Time used: 0.320 (sec). Leaf size: 27

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

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