67.8.6 problem 13.1 (f)

Internal problem ID [16501]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.1 (f)
Date solved : Thursday, October 02, 2025 at 01:35:37 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 10

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

\[ y = c_1 +\arctan \left (x \right ) c_2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \int _1^x\frac {c_1}{K[1]^2+1}dK[1]+c_2 \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = C_{1} + C_{2} \operatorname {atan}{\left (x \right )} \]