Internal problem ID [10916]
Internal file name [OUTPUT/10173_Sunday_December_31_2023_11_03_23_AM_45823441/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: 93.
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.
\[ \boxed {x y^{\prime \prime }+\left (x^{n}+1-n \right ) y^{\prime }+b \,x^{-1+2 n} y=0} \]
Maple trace Kovacic algorithm successful
`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 A Liouvillian solution exists Group is reducible or imprimitive <- Kovacics algorithm successful <- Equivalence, under non-integer power transformations successful`
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 53
dsolve(x*diff(y(x),x$2)+(x^n+1-n)*diff(y(x),x)+b*x^(2*n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{n}}{2 n}} \left (c_{1} \sinh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )+c_{2} \cosh \left (\frac {x^{n} \sqrt {\frac {-4 b +1}{n^{2}}}}{2}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.094 (sec). Leaf size: 53
DSolve[x*y''[x]+(x^n+1-n)*y'[x]+b*x^(2*n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\left (\sqrt {1-4 b}+1\right ) x^n}{2 n}} \left (c_2 e^{\frac {\sqrt {1-4 b} x^n}{n}}+c_1\right ) \]