Internal problem ID [10922]
Internal file name [OUTPUT/10179_Sunday_December_31_2023_11_03_36_AM_5864548/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 99.
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.
\[ \boxed {x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (c \,x^{-1+2 n}+d \,x^{n -1}\right ) y=0} \]
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: 0.172 (sec). Leaf size: 156
dsolve(x*diff(y(x),x$2)+(a*x^n+b)*diff(y(x),x)+(c*x^(2*n-1)+d*x^(n-1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{n} \left (\sqrt {a^{2}-4 c}+a \right )}{2 n}} \left (\operatorname {KummerU}\left (\frac {\left (b +n -1\right ) \sqrt {a^{2}-4 c}+a \left (b +n -1\right )-2 d}{2 \sqrt {a^{2}-4 c}\, n}, \frac {b +n -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (b +n -1\right ) \sqrt {a^{2}-4 c}+a \left (b +n -1\right )-2 d}{2 \sqrt {a^{2}-4 c}\, n}, \frac {b +n -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right ) c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.38 (sec). Leaf size: 255
DSolve[x*y''[x]+(a*x^n+b)*y'[x]+(c*x^(2*n-1)+d*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2^{\frac {b+n-1}{2 n}} x^{\frac {1}{2}-\frac {n}{2}} \left (x^n\right )^{\frac {n-1}{2 n}} e^{-\frac {\left (\sqrt {a^2-4 c}+a\right ) x^n}{2 n}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {(b+n-1) a^2+\sqrt {a^2-4 c} (b+n-1) a-2 \sqrt {a^2-4 c} d-4 c (b+n-1)}{2 \left (a^2-4 c\right ) n},\frac {b+n-1}{n},\frac {\sqrt {a^2-4 c} x^n}{n}\right )+c_2 L_{-\frac {(b+n-1) a^2+\sqrt {a^2-4 c} (b+n-1) a-2 \sqrt {a^2-4 c} d-4 c (b+n-1)}{2 \left (a^2-4 c\right ) n}}^{\frac {b-1}{n}}\left (\frac {\sqrt {a^2-4 c} x^n}{n}\right )\right ) \]