Internal problem ID [8723]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1143.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } a x +\left (x b +a \right ) y^{\prime }+c y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 67
dsolve(2*a*x*diff(diff(y(x),x),x)+(b*x+a)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {b x}{2 a}} \KummerM \left (\frac {-c +b}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) \sqrt {x}+c_{2} {\mathrm e}^{-\frac {b x}{2 a}} \KummerU \left (\frac {-c +b}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 70
DSolve[c*y[x] + (a + b*x)*y'[x] + 2*a*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt {x} e^{-\frac {b x}{2 a}} \left (c_1 \text {HypergeometricU}\left (1-\frac {c}{b},\frac {3}{2},\frac {b x}{2 a}\right )+c_2 L_{\frac {c}{b}-1}^{\frac {1}{2}}\left (\frac {b x}{2 a}\right )\right ) \\ \end{align*}