89.3.8 problem 8

Internal problem ID [24306]
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 : 8
Date solved : Thursday, October 02, 2025 at 10:12:52 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 31

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

\begin{align*} y &= \sqrt {{\mathrm e}^{-2 \arctan \left (x \right )} c_1 -1} \\ y &= -\sqrt {{\mathrm e}^{-2 \arctan \left (x \right )} c_1 -1} \\ \end{align*}

✓ Mathematica. Time used: 2.059 (sec). Leaf size: 59

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

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

✓ Sympy. Time used: 0.433 (sec). Leaf size: 32

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

\[ \left [ y{\left (x \right )} = - \sqrt {C_{1} e^{- 2 \operatorname {atan}{\left (x \right )}} - 1}, \ y{\left (x \right )} = \sqrt {C_{1} e^{- 2 \operatorname {atan}{\left (x \right )}} - 1}\right ] \]