60.3.43 problem 1044

Internal problem ID [11053]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1044
Date solved : Tuesday, January 28, 2025 at 05:40:43 PM
CAS classification : [_Hermite]

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

✓ Solution by Maple

Time used: 0.139 (sec). Leaf size: 35

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

\[ y = x \left (\operatorname {KummerM}\left (\frac {1}{2}+\frac {a}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {1}{2}+\frac {a}{2}, \frac {3}{2}, \frac {x^{2}}{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 39

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

\[ y(x)\to c_1 \operatorname {HermiteH}\left (-a,\frac {x}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a}{2},\frac {1}{2},\frac {x^2}{2}\right ) \]