78.4.12 problem 13

Internal problem ID [18064]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 13
Date solved : Monday, March 31, 2025 at 05:02:36 PM
CAS classification : [_exact, _rational, _Riccati]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 17

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

\[ y = \frac {i \tan \left (\frac {i x^{2}}{2}+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.134 (sec). Leaf size: 25

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

\[ y(x)\to -\frac {\tanh \left (\frac {1}{2} \left (x^2-2 i c_1\right )\right )}{x} \]

✓ Sympy. Time used: 0.270 (sec). Leaf size: 17

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

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