Riley hobson bence · Problem 15.37

Higher-order ordinary differential equations

Mathematics — Methods · higher-order-odes-isobaric

Problem

Consider the equation

x^p y'' + \frac{n+3-2p}{n-1}x^{p-1}y' + \left(\frac{p-2}{n-1}\right)^2 x^{p-2}y = y^n,

in which

p \neq 2

and

n > -1

but

n \neq 1

. For the boundary conditions

y(1) = 0

and

y'(1) = \lambda

, show that the solution is

y(x) = v(x)x^{(p-2)/(n-1)}

, where

v(x)

is given by

\int_0^{v(x)} \frac{dz}{\left[\lambda^2 + 2z^{n+1}/(n+1)\right]^{1/2}} = \ln x.