Internal problem ID [8596]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1016.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\left (a \,x^{2 c}+b \,x^{c -1}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.407 (sec). Leaf size: 95
dsolve(diff(diff(y(x),x),x)+(a*x^(2*c)+b*x^(c-1))*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \WhittakerM \left (-\frac {i b}{\sqrt {a}\, \left (2 c +2\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c +1}}{c +1}\right ) x^{-\frac {c}{2}}+c_{2} \WhittakerW \left (-\frac {i b}{\sqrt {a}\, \left (2 c +2\right )}, \frac {1}{2 c +2}, \frac {2 i \sqrt {a}\, x^{c +1}}{c +1}\right ) x^{-\frac {c}{2}} \]
✓ Solution by Mathematica
Time used: 0.073 (sec). Leaf size: 199
DSolve[(b*x^(-1 + c) + a*x^(2*c))*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{\frac {c}{2 c+2}} x^{-c/2} \left (x^{c+1}\right )^{\frac {c}{2 c+2}} e^{-\frac {\sqrt {a} x^{c+1}}{\sqrt {-(c+1)^2}}} \left (c_1 \text {HypergeometricU}\left (\frac {-\frac {b \sqrt {-(c+1)^2}}{\sqrt {a}}+c^2+c}{2 (c+1)^2},\frac {c}{c+1},\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-(c+1)^2}}\right )+c_2 \text {LaguerreL}\left (-\frac {-\frac {b \sqrt {-(c+1)^2}}{\sqrt {a}}+c^2+c}{2 (c+1)^2},-\frac {1}{c+1},\frac {2 \sqrt {a} x^{c+1}}{\sqrt {-(c+1)^2}}\right )\right ) \\ \end{align*}