56.4.64 problem 61
Internal
problem
ID
[8953]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
61
Date
solved
:
Thursday, March 13, 2025 at 06:58:29 PM
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: 0.035 (sec). Leaf size: 167
ode:=x/(1-x)*diff(diff(y(x),x),x)+y(x) = 1/(1-x);
dsolve(ode,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. Time used: 0.249 (sec). Leaf size: 136
ode=x/(1-x)*D[y[x],{x,2}]+y[x]==1/(1-x);
ic={};
DSolve[{ode,ic},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
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)