56.2.45 problem 44
Internal
problem
ID
[8849]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
44
Date
solved
:
Tuesday, January 28, 2025 at 03:24:09 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-y x -x^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 44
dsolve(diff(y(x),x$2)-x^2*diff(y(x),x)-x*y(x)-x^2=0,y(x), singsol=all)
\[
y = {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{2} +{\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right ) c_{1} -\frac {x}{2}
\]
✓ Solution by Mathematica
Time used: 0.318 (sec). Leaf size: 224
DSolve[D[y[x],{x,2}]-x^2*D[y[x],x]-x*y[x]-x^2==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to -\frac {e^{\frac {x^3}{6}} \left (12 \left (x^3\right )^{5/6} \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {7}{6}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (-\frac {2}{3};-\frac {1}{3};-\frac {x^3}{3}\right )+\sqrt [3]{2} 3^{2/3} \sqrt [6]{x^3} x^6 \operatorname {Gamma}\left (\frac {1}{6}\right ) \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right ) \, _1F_1\left (\frac {2}{3};\frac {7}{3};-\frac {x^3}{3}\right )-4 \operatorname {Gamma}\left (\frac {7}{6}\right ) \left (6 \sqrt [3]{2} 3^{2/3} c_1 x^{5/2} \operatorname {Gamma}\left (\frac {5}{6}\right ) \operatorname {BesselI}\left (-\frac {1}{6},\frac {x^3}{6}\right )+\operatorname {Gamma}\left (\frac {1}{6}\right ) \left (3 \left (x^3\right )^{5/6}+2 \sqrt [3]{-1} 3^{2/3} c_2 x^{5/2}\right ) \operatorname {BesselI}\left (\frac {1}{6},\frac {x^3}{6}\right )\right )\right )}{24\ 2^{2/3} 3^{5/6} x^2 \operatorname {Gamma}\left (\frac {7}{6}\right )}
\]