28.1.59 problem 60

Internal problem ID [4365]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 60
Date solved : Tuesday, March 04, 2025 at 06:33:13 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.132 (sec). Leaf size: 30

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

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

✓ Sympy. Time used: 0.961 (sec). Leaf size: 19

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

\[ C_{1} + x y{\left (x \right )} + \frac {y^{2}{\left (x \right )}}{2} - \log {\left (y{\left (x \right )} \right )} = 0 \]