50.4.64 problem 61
Internal
problem
ID
[10239]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
61
Date
solved
:
Friday, October 03, 2025 at 02:20:59 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} \frac {x y^{\prime \prime }}{1-x}+y&=\frac {1}{1-x} \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 157
ode:=x/(1-x)*diff(diff(y(x),x),x)+y(x) = 1/(1-x);
dsolve(ode,y(x), singsol=all);
\[
y = -\left (\left (-\operatorname {BesselI}\left (0, -x \right )-\operatorname {BesselI}\left (1, -x \right )\right ) \int \frac {\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )}{\operatorname {BesselI}\left (0, x\right ) \operatorname {BesselK}\left (1, -x \right ) x^{2}-\operatorname {BesselI}\left (1, x\right ) \operatorname {BesselK}\left (0, -x \right ) x^{2}+x +1}d x +\int \frac {\operatorname {BesselI}\left (0, x\right )-\operatorname {BesselI}\left (1, x\right )}{\operatorname {BesselI}\left (0, x\right ) \operatorname {BesselK}\left (1, -x \right ) x^{2}-\operatorname {BesselI}\left (1, x\right ) \operatorname {BesselK}\left (0, -x \right ) x^{2}+x +1}d x \left (\operatorname {BesselK}\left (0, -x \right )-\operatorname {BesselK}\left (1, -x \right )\right )-c_1 \operatorname {BesselK}\left (0, -x \right )+c_1 \operatorname {BesselK}\left (1, -x \right )-c_2 \operatorname {BesselI}\left (0, -x \right )-c_2 \operatorname {BesselI}\left (1, -x \right )\right ) x
\]
✓ Mathematica. Time used: 0.135 (sec). Leaf size: 119
ode=x/(1-x)*D[y[x],{x,2}]+y[x]==1/(1-x);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-x} x \left (\operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \int _1^x-2 \sqrt {\pi } (\operatorname {BesselI}(0,K[1])-\operatorname {BesselI}(1,K[1]))dK[1]+e^x (\operatorname {BesselI}(0,x)-\operatorname {BesselI}(1,x)) \int _1^x2 e^{-K[2]} \sqrt {\pi } \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 K[2]\right )dK[2]+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 ) \end{align*}
✗ Sympy
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)