2.2.21 Problem 22

Internal problem ID [9144]
Book : Second order enumerated odes
Section : section 2
Problem number : 22
Date solved : Sunday, February 23, 2025 at 05:33:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+y&=4 \cos \left (\ln \left (1+x \right )\right ) \end{align*}

Maple step by step solution

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`
 

Maple dsolve solution

Solving time : 0.077 (sec)
Leaf size : 280

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

\[ y = \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) c_{2} +\left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) c_{1} -80 \left (\int \frac {\left (i x -1\right ) \cos \left (\ln \left (x +1\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{\left (x^{2}+1\right ) \left (10 \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (\left (-1-i+\left (-1+i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+\left (1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right )+\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {3 i}{2}, \frac {3}{2}+\frac {i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (1+7 i+\left (7-i\right ) x \right )\right )}d x \right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )-80 \left (\int \frac {\operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \cos \left (\ln \left (x +1\right )\right ) \left (x +i\right )^{\frac {1}{2}+\frac {i}{2}}}{7 \left (\frac {10 \left (\left (1-i+\left (-1-i\right ) x \right ) \operatorname {hypergeom}\left (\left [1-i, 1+i\right ], \left [\frac {3}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )+\left (-1+i\right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )}{7}+\operatorname {hypergeom}\left (\left [\frac {3}{2}-\frac {3 i}{2}, \frac {3}{2}+\frac {i}{2}\right ], \left [\frac {5}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \left (-1+\frac {i}{7}+\left (\frac {1}{7}+i\right ) x \right ) \operatorname {hypergeom}\left (\left [i, -i\right ], \left [\frac {1}{2}+\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right )\right ) \left (x^{2}+1\right )}d x \right ) \left (x +i\right )^{\frac {1}{2}-\frac {i}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {i}{2}, \frac {1}{2}-\frac {3 i}{2}\right ], \left [\frac {3}{2}-\frac {i}{2}\right ], \frac {1}{2}-\frac {i x}{2}\right ) \]

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

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

Not solved

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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