56.4.66 problem 63
Internal
problem
ID
[8955]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
63
Date
solved
:
Sunday, March 30, 2025 at 01:56:23 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} \frac {x y^{\prime \prime }}{1-x}+y&=\cos \left (x \right ) \end{align*}
✓ Maple. Time used: 0.045 (sec). Leaf size: 167
ode:=x/(1-x)*diff(diff(y(x),x),x)+y(x) = cos(x);
dsolve(ode,y(x), singsol=all);
\[
y = -\left (\left (-\operatorname {BesselI}\left (0, -x \right )-\operatorname {BesselI}\left (1, -x \right )\right ) \int \frac {\left (-\operatorname {BesselK}\left (0, -x \right )+\operatorname {BesselK}\left (1, -x \right )\right ) \cos \left (x \right ) \left (-1+x \right )}{\operatorname {BesselK}\left (1, -x \right ) \operatorname {BesselI}\left (0, x\right ) x^{2}-\operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right ) x^{2}+x +1}d x +\int \frac {\left (\operatorname {BesselI}\left (0, x\right )-\operatorname {BesselI}\left (1, x\right )\right ) \cos \left (x \right ) \left (-1+x \right )}{\operatorname {BesselK}\left (1, -x \right ) \operatorname {BesselI}\left (0, x\right ) x^{2}-\operatorname {BesselK}\left (0, -x \right ) \operatorname {BesselI}\left (1, x\right ) x^{2}+x +1}d x \left (-\operatorname {BesselK}\left (0, -x \right )+\operatorname {BesselK}\left (1, -x \right )\right )-c_1 \operatorname {BesselK}\left (1, -x \right )+c_1 \operatorname {BesselK}\left (0, -x \right )-c_2 \operatorname {BesselI}\left (0, -x \right )-c_2 \operatorname {BesselI}\left (1, -x \right )\right ) x
\]
✓ Mathematica. Time used: 8.139 (sec). Leaf size: 133
ode=x/(1-x)*D[y[x],{x,2}]+y[x]==Cos[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-x} x \left (\operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right ) \int _1^x2 \sqrt {\pi } (\operatorname {BesselI}(0,K[1])-\operatorname {BesselI}(1,K[1])) \cos (K[1]) (K[1]-1)dK[1]+e^x (\operatorname {BesselI}(0,x)-\operatorname {BesselI}(1,x)) \int _1^x-2 e^{-K[2]} \sqrt {\pi } \cos (K[2]) \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 K[2]\right ) (K[2]-1)dK[2]+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2},2,2 x\right )+c_2 e^x \operatorname {BesselI}(0,x)-c_2 e^x \operatorname {BesselI}(1,x)\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), (x, 2))/(1 - x) + y(x) - cos(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve x*Derivative(y(x), (x, 2))/(1 - x) + y(x) - cos(x)