28.6 problem 66

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.

\[ \boxed {x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y=0} \]

28.6.1 Solving as second order ode lagrange adjoint equation method ode

In normal form the ode \begin {align*} x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y = 0 \tag {1} \end {align*}

Becomes \begin {align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=r \left (x \right ) \tag {2} \end {align*}

Where \begin {align*} p \left (x \right )&=\frac {1-3 n}{x}\\ q \left (x \right )&=-x^{-2+2 n} a^{2} n^{2}\\ r \left (x \right )&=0 \end {align*}

The Lagrange adjoint ode is given by \begin {align*} \xi ^{''}-(\xi \, p)'+\xi q &= 0\\ \xi ^{''}-\left (\frac {\left (1-3 n \right ) \xi \left (x \right )}{x}\right )' + \left (-x^{-2+2 n} a^{2} n^{2} \xi \left (x \right )\right ) &= 0\\ \xi ^{\prime \prime }\left (x \right )-\frac {\left (1-3 n \right ) \xi ^{\prime }\left (x \right )}{x}+\left (\frac {1-3 n}{x^{2}}-x^{-2+2 n} a^{2} n^{2}\right ) \xi \left (x \right )&= 0 \end {align*}

Which is solved for \(\xi (x)\).

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]
 

\[ y(x)\to \left (c_1-\frac {3}{8} i a c_2 \sqrt {x^{2 n}}\right ) \cosh \left (a \sqrt {x^{2 n}}\right )+\frac {1}{8} \left (3 i c_2-8 a c_1 \sqrt {x^{2 n}}\right ) \sinh \left (a \sqrt {x^{2 n}}\right ) \]