89.5.16 problem 16

Internal problem ID [24366]
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 : 16
Date solved : Thursday, October 02, 2025 at 10:22:16 PM
CAS classification : [_linear]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime }+2 y&=\left (x^{2}+1\right )^{3} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 21

ode:=x*(x^2+1)*diff(y(x),x)+2*y(x) = (x^2+1)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\frac {\left (x^{2}+1\right )^{2}}{4}+c_1 \right ) \left (1+\frac {1}{x^{2}}\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 30

ode=x*(x^2+1)*D[y[x],x]+2*y[x]==(x^2+1)^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\left (x^2+1\right ) \left (x^4+2 x^2+4 c_1\right )}{4 x^2} \end{align*}

✓ Sympy. Time used: 0.245 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + 1)*Derivative(y(x), x) - (x**2 + 1)**3 + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {C_{1}}{x^{2}} + \frac {x^{4}}{4} + \frac {3 x^{2}}{4} + \frac {1}{2} \]