15.18.8 problem 8

Internal problem ID [3251]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 8
Date solved : Sunday, March 30, 2025 at 01:25:06 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 15

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

\[ y = -x +\left (c_1 +1\right ) \arctan \left (x \right )+c_2 \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 18

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

\[ y(x)\to (1+c_1) \arctan (x)-x+c_2 \]

✓ Sympy. Time used: 0.572 (sec). Leaf size: 19

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

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