68.2.1 problem Problem 3.7(a)

Internal problem ID [14146]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 3 Bessel functions. Problems page 89
Problem number : Problem 3.7(a)
Date solved : Tuesday, January 28, 2025 at 06:15:54 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 27

dsolve(diff(y(x),x$2)-x^2*y(x)=0,y(x), singsol=all)
 

\[ y = \left (\operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{1} +\operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 37

DSolve[D[y[x],{x,2}]-x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} x\right )+c_1 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} x\right ) \]