56.2.38 problem 37
Internal
problem
ID
[8842]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
37
Date
solved
:
Monday, January 27, 2025 at 05:04:47 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }-x^{2} y-x^{4}+2&=0 \end{align*}
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 34
dsolve(diff(y(x),x$2)-x^2*y(x)-x^4+2=0,y(x), singsol=all)
\[
y = \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{2} +\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{1} -x^{2}
\]
✓ Solution by Mathematica
Time used: 4.495 (sec). Leaf size: 217
DSolve[D[y[x],{x,2}]-x^2*y[x]-x^4+2==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} x\right ) \left (\int _1^x-\frac {\left (K[1]^4-2\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} K[1]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[1]\right )+\operatorname {HermiteH}\left (-\frac {1}{2},K[1]\right ) \left (-i \operatorname {HermiteH}\left (\frac {1}{2},i K[1]\right )-2 \operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) K[1]\right )\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} x\right ) \left (\int _1^x\frac {\left (K[2]^4-2\right ) \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} K[2]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},i K[2]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[2]\right )+\operatorname {HermiteH}\left (-\frac {1}{2},K[2]\right ) \left (-i \operatorname {HermiteH}\left (\frac {1}{2},i K[2]\right )-2 \operatorname {HermiteH}\left (-\frac {1}{2},i K[2]\right ) K[2]\right )\right )}dK[2]+c_2\right )
\]