Internal problem ID [10889]
Internal file name [OUTPUT/10146_Sunday_December_24_2023_05_15_09_PM_18821174/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: 66.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Emden, _Fowler]]
Unable to solve or complete the solution.
\[ \boxed {x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y=0} \]
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: 0.094 (sec). Leaf size: 62
dsolve(x*diff(y(x),x$2)+(1-3*n)*diff(y(x),x)-a^2*n^2*x^(2*n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{-a \,x^{n}} \left (a \,x^{n}+x^{-n} \sqrt {x^{2 n}}\right )-{\mathrm e}^{a \,x^{n}} c_{1} \left (a \,x^{n}-x^{-n} \sqrt {x^{2 n}}\right ) \]
✓ Solution by Mathematica
Time used: 0.196 (sec). Leaf size: 77
DSolve[x*y''[x]+(1-3*n)*y'[x]-a^2*n^2*x^(2*n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (c_1-\frac {3}{8} i a c_2 \sqrt {x^{2 n}}\right ) \cosh \left (a \sqrt {x^{2 n}}\right )+\frac {1}{8} \left (3 i c_2-8 a c_1 \sqrt {x^{2 n}}\right ) \sinh \left (a \sqrt {x^{2 n}}\right ) \]