88.12.17 problem 19

Internal problem ID [24073]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Miscellaneous Exercises at page 55
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:57:02 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.295 (sec). Leaf size: 41

ode=(x*y[x]-x)==(x*y[x]^2+x-y[x]^2-1)*D[y[x],x]; 
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 ][x+\log (x-1)+c_1]\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.328 (sec). Leaf size: 26

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

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