75.20.7 problem 642

Internal problem ID [17070]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.5 Linear equations with variable coefficients. The Lagrange method. Exercises page 148
Problem number : 642
Date solved : Monday, March 31, 2025 at 03:39:55 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

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

\[ y = \sqrt {x^{2}+1}\, c_2 +c_1 x -1 \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 26

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

\[ y(x)\to c_1 \sqrt {x^2+1}+i c_2 x-1 \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*Derivative(y(x), (x, 2)) + y(x) - Derivative(y(x), (x, 2)) + 1)/x cannot be solved by the factorable group method