60.3.83 problem 1087

Internal problem ID [11093]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1087
Date solved : Tuesday, January 28, 2025 at 05:41:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y^{\prime \prime }-\left (x^{2}+a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.216 (sec). Leaf size: 33

dsolve(4*diff(diff(y(x),x),x)-(x^2+a)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} \operatorname {WhittakerW}\left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )+c_{1} \operatorname {WhittakerM}\left (-\frac {a}{8}, \frac {1}{4}, \frac {x^{2}}{2}\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 36

DSolve[(-a - x^2)*y[x] + 4*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {ParabolicCylinderD}\left (\frac {1}{4} (-a-2),x\right )+c_2 \operatorname {ParabolicCylinderD}\left (\frac {a-2}{4},i x\right ) \]