56.2.50 problem 49
Internal
problem
ID
[8854]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
49
Date
solved
:
Sunday, March 30, 2025 at 01:42:58 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-x^{3} y-x^{4}-\frac {1}{x}&=0 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 26
ode:=diff(diff(y(x),x),x)-1/x*diff(y(x),x)-x^3*y(x)-x^4-1/x = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x \left (-1+c_2 \operatorname {BesselI}\left (\frac {2}{5}, \frac {2 x^{{5}/{2}}}{5}\right )+c_1 \operatorname {BesselK}\left (\frac {2}{5}, \frac {2 x^{{5}/{2}}}{5}\right )\right )
\]
✓ Mathematica. Time used: 0.334 (sec). Leaf size: 316
ode=D[y[x],{x,2}]-1/x*D[y[x],x]-x^3*y[x]-x^4-1/x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {\frac {5 \left (x^{5/2}\right )^{13/5} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {4}{5};\frac {3}{5},\frac {9}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {9}{5}\right )}-\frac {\sqrt [5]{5} \left (x^{5/2}\right )^{7/5} \operatorname {Gamma}\left (\frac {1}{5}\right ) \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {1}{5};\frac {6}{5},\frac {7}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {6}{5}\right )}+\frac {5 \left (x^{5/2}\right )^{3/5} \operatorname {Gamma}\left (-\frac {1}{5}\right ) \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (-\frac {1}{5};\frac {3}{5},\frac {4}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {4}{5}\right )}+\sqrt [5]{5} x^{5/2} \left (-\frac {x^5 \left (x^{5/2}\right )^{2/5} \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right ) \, _1F_2\left (\frac {6}{5};\frac {7}{5},\frac {11}{5};\frac {x^5}{25}\right )}{\operatorname {Gamma}\left (\frac {11}{5}\right )}+10 \left (c_1 \operatorname {Gamma}\left (\frac {3}{5}\right ) \operatorname {BesselI}\left (-\frac {2}{5},\frac {2 x^{5/2}}{5}\right )+(-1)^{2/5} c_2 \operatorname {Gamma}\left (\frac {7}{5}\right ) \operatorname {BesselI}\left (\frac {2}{5},\frac {2 x^{5/2}}{5}\right )\right )\right )}{10\ 5^{3/5} x^{3/2}}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**4 - x**3*y(x) + Derivative(y(x), (x, 2)) - Derivative(y(x), x)/x - 1/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE x**5 + x**4*y(x) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + 1 cannot be solved by the factorable group method