Internal problem ID [8719]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1139.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {16 x y^{\prime \prime }+8 y^{\prime }-\left (x +a \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 43
dsolve(16*x*diff(diff(y(x),x),x)+8*diff(y(x),x)-(x+a)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x}\, {\mathrm e}^{-\frac {x}{4}} \KummerM \left (\frac {a}{8}+\frac {3}{4}, \frac {3}{2}, \frac {x}{2}\right )+c_{2} \sqrt {x}\, {\mathrm e}^{-\frac {x}{4}} \KummerU \left (\frac {a}{8}+\frac {3}{4}, \frac {3}{2}, \frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 59
DSolve[(-a - x)*y[x] + 8*y'[x] + 16*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x/4} \sqrt {x} \left (c_1 \text {HypergeometricU}\left (\frac {a+6}{8},\frac {3}{2},\frac {x}{2}\right )+c_2 L_{\frac {1}{8} (-a-6)}^{\frac {1}{2}}\left (\frac {x}{2}\right )\right ) \\ \end{align*}