60.3.12 problem 1012

Internal problem ID [11022]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1012
Date solved : Tuesday, January 28, 2025 at 05:40:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (x^{2}+a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.250 (sec). Leaf size: 29

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

\[ y = \frac {c_{1} \operatorname {WhittakerM}\left (-\frac {a}{4}, \frac {1}{4}, x^{2}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a}{4}, \frac {1}{4}, x^{2}\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 47

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

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