Internal problem ID [8638]
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]]
\[ \boxed {y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y=0} \]
✓ Solution by Maple
Time used: 0.359 (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)
\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-\frac {a \,x^{q}}{q}} \operatorname {KummerM}\left (\frac {a q -b}{a q}, \frac {q +1}{q}, \frac {a \,x^{q}}{q}\right )+c_{2} x \,{\mathrm e}^{-\frac {a \,x^{q}}{q}} \operatorname {KummerU}\left (\frac {a q -b}{a q}, \frac {q +1}{q}, \frac {a \,x^{q}}{q}\right ) \]
✓ Solution by Mathematica
Time used: 0.019 (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}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{a q},1+\frac {1}{q},-\frac {a x^q}{q}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{a q},\frac {q-1}{q},-\frac {a x^q}{q}\right ) \\ \end{align*}