2.2.21 Problem 22

Internal problem ID [9144]
Book : Second order enumerated odes
Section : section 2
Problem number : 22
Date solved : Sunday, March 30, 2025 at 02:23:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

✓ Maple. Time used: 0.191 (sec). Leaf size: 280

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

\begin{align*} \text {Solution too large to show}\end{align*}

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
trying a double symmetry of the form [xi=0, eta=F(x)] 
-> Try solving first the homogeneous part of the ODE 
   checking if the LODE has constant coefficients 
   checking if the LODE is of Euler type 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Trying a Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         <- heuristic approach successful 
      <- hypergeometric successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful
 

✗ Mathematica

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

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x + 1)*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)) + y(x) - 4*cos(log(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) + 4*cos(log(x + 1)) - Derivative(y(x), (x, 2)))/(x + 1) cannot be solved by the factorable group method