26.6.44 problem Exercise 12.44, page 103

Internal problem ID [7044]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.44, page 103
Date solved : Tuesday, September 30, 2025 at 04:17:15 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 25

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

\[ \text {Solve}\left [x y(x)-\frac {1}{x y(x)}-2 \log (y(x))=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.729 (sec). Leaf size: 26

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

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