56.2.41 problem 40
Internal
problem
ID
[8845]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
2.0
Problem
number
:
40
Date
solved
:
Tuesday, January 28, 2025 at 03:18:07 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-x^{3} y-x^{4}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 32
dsolve(diff(y(x),x$2)-x^3*y(x)-x^4=0,y(x), singsol=all)
\[
y = \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{5}, \frac {2 x^{{5}/{2}}}{5}\right ) c_{2} +\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{5}, \frac {2 x^{{5}/{2}}}{5}\right ) c_{1} -x
\]
✓ Solution by Mathematica
Time used: 0.187 (sec). Leaf size: 219
DSolve[D[y[x],{x,2}]-x^3*y[x]-x^4==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\sqrt [5]{-1} \operatorname {Gamma}\left (\frac {6}{5}\right ) \left (-5^{2/5} \sqrt [5]{x^{5/2}} x^{15/2} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {Hypergeometric0F1Regularized}\left (\frac {11}{5},\frac {x^5}{25}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{4/5} \left (x^{5/2}\right )^{4/5} \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{3/5} x^5 \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {6}{5},\frac {2 x^{5/2}}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )\right )}{25 x^{3/2} \text {Root}\left [25 \text {$\#$1}^5+1\&,5\right ]}+\frac {c_1 \sqrt {x} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )}{\sqrt [5]{5}}+\sqrt [5]{-\frac {1}{5}} c_2 \sqrt {x} \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )
\]