89.3.9 problem 9

Internal problem ID [24307]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 34
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:12:55 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact, _rational, _Bernoulli]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 34

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

\begin{align*} y &= \frac {\sqrt {-2 x +c_1}}{x +1} \\ y &= -\frac {\sqrt {-2 x +c_1}}{x +1} \\ \end{align*}

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to -\frac {\sqrt {-2 x+c_1}}{\sqrt {(x+1)^2}}\\ y(x)&\to \frac {\sqrt {-2 x+c_1}}{\sqrt {(x+1)^2}} \end{align*}

✓ Sympy. Time used: 0.322 (sec). Leaf size: 29

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

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