2.4.64 Problem 61

Internal problem ID [8953]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 61
Date solved : Sunday, February 23, 2025 at 05:33:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

Solve

\begin{align*} \frac {x y^{\prime \prime }}{1-x}+y&=\frac {1}{1-x} \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 successful 
   <- special function solution successful 
<- solving first the homogeneous part of the ODE successful`
 

Maple dsolve solution

Solving time : 0.035 (sec)
Leaf size : 167

dsolve(x/(1-x)*diff(diff(y(x),x),x)+y(x) = 1/(1-x),y(x),singsol=all)
 

\[ y = -x \left (\left (\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )\right ) \left (\int \frac {-\operatorname {BesselI}\left (0, -x \right )-\operatorname {BesselI}\left (1, -x \right )}{x \left (\operatorname {BesselI}\left (0, x\right ) \left (x +1\right ) \operatorname {BesselK}\left (1, -x \right )+1-\left (x +1\right ) \operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right )\right )}d x \right )+\left (-\operatorname {BesselI}\left (0, -x \right )-\operatorname {BesselI}\left (1, -x \right )\right ) \left (\int \frac {-\operatorname {BesselK}\left (0, -x \right )+\operatorname {BesselK}\left (1, -x \right )}{x \left (\operatorname {BesselI}\left (0, x\right ) \left (x +1\right ) \operatorname {BesselK}\left (1, -x \right )+1-\left (x +1\right ) \operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right )\right )}d x \right )-\operatorname {BesselK}\left (0, -x \right ) c_{1} +\operatorname {BesselK}\left (1, -x \right ) c_{1} -\operatorname {BesselI}\left (0, -x \right ) c_{2} -\operatorname {BesselI}\left (1, -x \right ) c_{2} \right ) \]

✓Mathematica DSolve solution

Solving time : 0.249 (sec)
Leaf size : 136

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

\[ y(x)\to e^{-x} x \left (e^x (\operatorname {BesselI}(0,x)-\operatorname {BesselI}(1,x)) \int _1^x2 e^{-K[1]} \sqrt {\pi } \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 K[1]\right )dK[1]-2 \sqrt {\pi } x \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \, _1F_2\left (\frac {1}{2};1,\frac {3}{2};\frac {x^2}{4}\right )+2 \sqrt {\pi } \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \operatorname {BesselI}(0,x)+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right )+c_2 e^x \operatorname {BesselI}(0,x)-c_2 e^x \operatorname {BesselI}(1,x)\right ) \]

✗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*Derivative(y(x), (x, 2))/(1 - x) + y(x) - 1/(1 - x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve x*Derivative(y(x), (x, 2))/(1 - x) + y(x) - 1/(1 - x)