28.1.49 problem 50

Internal problem ID [4355]
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 : 50
Date solved : Tuesday, March 04, 2025 at 06:32:48 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 138

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

\[ -\frac {\left (\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2} y \left (x \right )^{2}}{\left (x -1\right ) \left (y \left (x \right )^{2}-x +1\right )}}\, \left (x -1\right )}{\sqrt {\frac {x -1}{x -1-y \left (x \right )^{2}}}\, x}\right )-c_{1} \right ) \sqrt {\frac {x^{2} y \left (x \right )^{2}}{\left (x -1\right ) \left (y \left (x \right )^{2}-x +1\right )}}-\frac {\sqrt {\frac {2 x -2}{x -1-y \left (x \right )^{2}}}\, \sqrt {2}}{2}}{\sqrt {\frac {x^{2} y \left (x \right )^{2}}{\left (x -1\right ) \left (y \left (x \right )^{2}-x +1\right )}}} = 0 \]

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 34

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

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

✓ Sympy. Time used: 1.200 (sec). Leaf size: 19

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

\[ C_{1} + \operatorname {atan}{\left (y{\left (x \right )} \right )} + \frac {1}{y{\left (x \right )}} - \frac {1}{x y{\left (x \right )}} = 0 \]