Internal problem ID [8640]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1060.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.057 (sec). Leaf size: 91
dsolve(diff(diff(y(x),x),x)+a*x^(q-1)*diff(y(x),x)+b*x^(q-2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x^{q} a}{q}} \KummerM \left (\frac {a q -b}{a q}, \frac {q +1}{q}, \frac {x^{q} a}{q}\right ) x +c_{2} {\mathrm e}^{-\frac {x^{q} a}{q}} \KummerU \left (\frac {a q -b}{a q}, \frac {q +1}{q}, \frac {x^{q} a}{q}\right ) x \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 81
DSolve[b*x^(-2 + q)*y[x] + a*x^(-1 + q)*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 q^{-1/q} a^{\frac {1}{q}} \left (x^q\right )^{\frac {1}{q}} \, _1F_1\left (\frac {a+b}{a q};1+\frac {1}{q};-\frac {a x^q}{q}\right )+c_1 \, _1F_1\left (\frac {b}{a q};\frac {q-1}{q};-\frac {a x^q}{q}\right ) \\ \end{align*}