56.2.48 problem 47
Internal
problem
ID
[8852]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
47
Date
solved
:
Thursday, March 13, 2025 at 06:16:42 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-x y-x^{2}-\frac {1}{x}&=0 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 26
ode:=diff(diff(y(x),x),x)-1/x*diff(y(x),x)-x*y(x)-x^2-1/x = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x \left (-1+\operatorname {BesselI}\left (\frac {2}{3}, \frac {2 x^{{3}/{2}}}{3}\right ) c_{2} +\operatorname {BesselK}\left (\frac {2}{3}, \frac {2 x^{{3}/{2}}}{3}\right ) c_{1} \right )
\]
✓ Mathematica. Time used: 0.449 (sec). Leaf size: 253
ode=D[y[x],{x,2}]-1/x*D[y[x],x]-x*y[x]-x^2-1/x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {\frac {3 \sqrt [6]{3} \pi \operatorname {Gamma}\left (-\frac {1}{3}\right ) \left (3 \operatorname {AiryAiPrime}(x)+\sqrt {3} \operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (-\frac {1}{3};\frac {1}{3},\frac {2}{3};\frac {x^3}{9}\right )}{x \operatorname {Gamma}\left (\frac {2}{3}\right )}+\frac {\frac {\sqrt [3]{3} \pi x \operatorname {Gamma}\left (\frac {1}{3}\right )^2 \left (\sqrt {3} \operatorname {AiryAiPrime}(x)-\operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {1}{3};\frac {4}{3},\frac {5}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {4}{3}\right )}+\frac {\sqrt [3]{3} \pi x^4 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {4}{3}\right ) \left (\sqrt {3} \operatorname {AiryAiPrime}(x)-\operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {4}{3};\frac {5}{3},\frac {7}{3};\frac {x^3}{9}\right )}{\operatorname {Gamma}\left (\frac {7}{3}\right )}+3 \sqrt [6]{3} \pi x^2 \operatorname {Gamma}\left (\frac {2}{3}\right ) \left (3 \operatorname {AiryAiPrime}(x)+\sqrt {3} \operatorname {AiryBiPrime}(x)\right ) \, _1F_2\left (\frac {2}{3};\frac {1}{3},\frac {5}{3};\frac {x^3}{9}\right )+27 \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Gamma}\left (\frac {5}{3}\right ) (c_1 \operatorname {AiryAiPrime}(x)+c_2 \operatorname {AiryBiPrime}(x))}{\operatorname {Gamma}\left (\frac {5}{3}\right )}}{27 \operatorname {Gamma}\left (\frac {1}{3}\right )}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2 - x*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**3 + x**2*y(x) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + 1 cannot be solved by the factorable group method