89.5.8 problem 8

Internal problem ID [24358]
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 43
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:22:02 PM
CAS classification : [_linear]

\begin{align*} 2 y x +x^{2}+x^{4}-\left (x^{2}+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=2*x*y(x)+x^2+x^4-(x^2+1)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x -\arctan \left (x \right )+c_1 \right ) \left (x^{2}+1\right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 19

ode=(2*x*y[x] +x^2 +x^4)-(1+x^2 )*D[y[x],x]== 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.290 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4 + x**2 + 2*x*y(x) - (x**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} x^{2} + C_{1} + x^{3} + \frac {i x^{2} \log {\left (x - i \right )}}{2} - \frac {i x^{2} \log {\left (x + i \right )}}{2} + x + \frac {i \log {\left (x - i \right )}}{2} - \frac {i \log {\left (x + i \right )}}{2} \]