60.3.40 problem 1041

Internal problem ID [11050]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1041
Date solved : Tuesday, January 28, 2025 at 05:40:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x y^{\prime }+\left (n +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.145 (sec). Leaf size: 41

dsolve(diff(diff(y(x),x),x)+x*diff(y(x),x)+(n+1)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 47

DSolve[(1 + n)*y[x] + x*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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