Internal problem ID [11062]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {y^{\prime \prime }-x^{2} y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 29
dsolve(diff(y(x),x$2)-x^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 37
DSolve[y''[x]-x^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}