Internal problem ID [8768]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1188.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.035 (sec). Leaf size: 135
dsolve(x^2*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \KummerM \left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right )+c_{2} x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \KummerU \left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right ) \]
✓ Solution by Mathematica
Time used: 0.112 (sec). Leaf size: 223
DSolve[c*y[x] + (b + a*x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -i^{-\sqrt {(a-1)^2-4 c}+a+1} b^{\frac {1}{2} \left (-\sqrt {(a-1)^2-4 c}+a-1\right )} \left (\frac {1}{x}\right )^{\frac {1}{2} \left (-\sqrt {(a-1)^2-4 c}+a-1\right )} \left (c_2 i^{2 \sqrt {(a-1)^2-4 c}} b^{\sqrt {(a-1)^2-4 c}} \left (\frac {1}{x}\right )^{\sqrt {(a-1)^2-4 c}} \, _1F_1\left (\frac {1}{2} \left (a+\sqrt {(a-1)^2-4 c}-1\right );\sqrt {(a-1)^2-4 c}+1;\frac {b}{x}\right )+c_1 \, _1F_1\left (\frac {1}{2} \left (a-\sqrt {(a-1)^2-4 c}-1\right );1-\sqrt {(a-1)^2-4 c};\frac {b}{x}\right )\right ) \\ \end{align*}