56.2.45 problem 44
Internal
problem
ID
[8849]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
44
Date
solved
:
Sunday, March 30, 2025 at 01:42:38 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-x y-x^{2}&=0 \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 44
ode:=diff(diff(y(x),x),x)-x^2*diff(y(x),x)-x*y(x)-x^2 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_2 +{\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_1 -\frac {x}{2}
\]
✓ Mathematica. Time used: 0.318 (sec). Leaf size: 224
ode=D[y[x],{x,2}]-x^2*D[y[x],x]-x*y[x]-x^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to -\frac {e^{\frac {x^3}{6}} \left (12 \left (x^3\right )^{5/6} \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {7}{6}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (-\frac {2}{3};-\frac {1}{3};-\frac {x^3}{3}\right )+\sqrt [3]{2} 3^{2/3} \sqrt [6]{x^3} x^6 \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (\frac {2}{3};\frac {7}{3};-\frac {x^3}{3}\right )-4 \operatorname {Gamma}\left (\frac {7}{6}\right ) \left (6 \sqrt [3]{2} 3^{2/3} c_1 x^{5/2} \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right )+\operatorname {Gamma}\left (\frac {1}{6}\right ) \left (3 \left (x^3\right )^{5/6}+2 \sqrt [3]{-1} 3^{2/3} c_2 x^{5/2}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right )\right )\right )}{24\ 2^{2/3} 3^{5/6} x^2 \operatorname {Gamma}\left (\frac {7}{6}\right )}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2*Derivative(y(x), x) - x**2 - x*y(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) + 1 + y(x)/x - Derivative(y(x), (x, 2))/x**2 cannot be solved by the factorable group method