Internal problem ID [9392]
Internal file name [OUTPUT/8329_Monday_June_06_2022_02_44_45_AM_60643184/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+a \,x^{q -1} y^{\prime }+b \,x^{q -2} y=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.437 (sec). Leaf size: 71
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)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a \,x^{q}}{q}} x \left (\operatorname {KummerM}\left (1-\frac {b}{a q}, 1+\frac {1}{q}, \frac {a \,x^{q}}{q}\right ) c_{1} +\operatorname {KummerU}\left (1-\frac {b}{a q}, 1+\frac {1}{q}, \frac {a \,x^{q}}{q}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.072 (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]
\[ y(x)\to c_2 q^{-1/q} a^{\frac {1}{q}} \left (x^q\right )^{\frac {1}{q}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{a q},1+\frac {1}{q},-\frac {a x^q}{q}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{a q},\frac {q-1}{q},-\frac {a x^q}{q}\right ) \]