78.24.3 problem 4
Internal
problem
ID
[18460]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
9.
Laplace
transforms.
Section
51.
Derivatives
and
Integrals
of
Laplace
Transforms.
Problems
at
page
467
Problem
number
:
4
Date
solved
:
Tuesday, January 28, 2025 at 08:28:39 PM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} y^{\prime \prime }+x^{2} y&=0 \end{align*}
Using Laplace method With initial conditions
\begin{align*} y \left (0\right )&=y_{0}\\ y^{\prime }\left (0\right )&=\operatorname {yd}_{0} \end{align*}
✓ Solution by Maple
Time used: 0.406 (sec). Leaf size: 130
dsolve([diff(y(x),x$2)+x^2*y(x)=0,y(0) = y__0, D(y)(0) = yd__0],y(x), singsol=all)
\[
y = \frac {15 \pi ^{{3}/{2}} \operatorname {yd}_{0} \mathcal {L}^{-1}\left (\frac {\operatorname {StruveH}\left (\frac {3}{4}, \frac {\textit {\_s1}^{2}}{2}\right )}{\textit {\_s1}^{{3}/{2}}}, \textit {\_s1} , x\right )+5 \Gamma \left (\frac {3}{4}\right ) \left (\sqrt {2}\, \sqrt {\pi }\, y_{0} \Gamma \left (\frac {3}{4}\right )+4 c_{1} \right ) \mathcal {L}^{-1}\left (\sqrt {\textit {\_s1}}\, \operatorname {BesselY}\left (\frac {1}{4}, \frac {\textit {\_s1}^{2}}{2}\right ), \textit {\_s1} , x\right )+10 \sqrt {x}\, \sqrt {2}\, y_{0} \operatorname {BesselJ}\left (-\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}-5 \pi ^{{3}/{2}} \operatorname {yd}_{0} \mathcal {L}^{-1}\left (\sqrt {\textit {\_s1}}\, \operatorname {StruveH}\left (\frac {7}{4}, \frac {\textit {\_s1}^{2}}{2}\right ), \textit {\_s1} , x\right )+4 \delta \left (2, x\right ) \operatorname {yd}_{0} \Gamma \left (\frac {3}{4}\right )+20 c_{2} \mathcal {L}^{-1}\left (\sqrt {\textit {\_s1}}\, \operatorname {BesselJ}\left (\frac {1}{4}, \frac {\textit {\_s1}^{2}}{2}\right ), \textit {\_s1} , x\right ) \Gamma \left (\frac {3}{4}\right )}{20 \Gamma \left (\frac {3}{4}\right )}
\]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 83
DSolve[{D[y[x],{x,2}]+x^2*y[x]==0,{y[0]==y0,Derivative[1][y][0] == yd0}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(-1+i) x\right ) \left (\sqrt {2} \text {yd0} \operatorname {Gamma}\left (\frac {1}{4}\right )+(2+2 i) \text {y0} \operatorname {Gamma}\left (\frac {3}{4}\right )\right )-\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},(1+i) x\right ) \left (\sqrt {2} \text {yd0} \operatorname {Gamma}\left (\frac {1}{4}\right )-(2-2 i) \text {y0} \operatorname {Gamma}\left (\frac {3}{4}\right )\right )}{2\ 2^{3/4} \sqrt {\pi }}
\]