Internal problem ID [9474]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1144.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 a x y^{\prime \prime }+\left (b x +3 a \right ) y^{\prime }+y c=0} \]
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 153
dsolve(2*a*x*diff(diff(y(x),x),x)+(b*x+3*a)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-c_{1} c \left (a \left (b +4 c \right )-b^{2} x \right ) \operatorname {KummerM}\left (\frac {b -2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right )+c_{2} b \left (a \left (b +4 c \right )-b^{2} x \right ) \operatorname {KummerU}\left (\frac {b -2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right )+2 a \left (c c_{1} \left (c +b \right ) \operatorname {KummerM}\left (-\frac {b +2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right )+\operatorname {KummerU}\left (-\frac {b +2 c}{2 b}, \frac {3}{2}, \frac {b x}{2 a}\right ) b^{2} c_{2} \right )\right ) {\mathrm e}^{-\frac {b x}{2 a}}}{a \left (b -2 c \right ) c} \]
✓ Solution by Mathematica
Time used: 0.051 (sec). Leaf size: 69
DSolve[c*y[x] + (3*a + b*x)*y'[x] + 2*a*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {b x}{2 a}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {3}{2}-\frac {c}{b},\frac {3}{2},\frac {b x}{2 a}\right )+c_2 L_{\frac {c}{b}-\frac {3}{2}}^{\frac {1}{2}}\left (\frac {b x}{2 a}\right )\right ) \]