- #1
TheFerruccio
- 220
- 0
Homework Statement
Find the general solution to the boundary value problem.
Homework Equations
[tex]
(xy')' + \lambda x^{-1}y = 0
[/tex]
[tex]y(1) = 0[/tex]
[tex]y(e) = 0[/tex]
use [tex]x = e^t[/tex]
The Attempt at a Solution
[tex]x = e^t[/tex] so [tex]\frac{dx}{dt} = e^t[/tex]
using chain rule:
[tex]y' = e^{-t}\frac{dy}{dt}[/tex]
Substituting this in:
[tex]\frac{d}{dx}(e^t(e^{-t}\frac{dy}{dt})) + \lambda e^{-t}y = 0[/tex]
[tex]\frac{d}{dx}(\frac{dy}{dt}) = \lambda e^{-t}y = 0[/tex]
[tex]\frac{d}{dx}(y'e^t) + \lambda e^{-t}y = 0[/tex]
From this point, I feel like I am going in circles. I want to get everything in equal powers of [tex]e[/tex], so I can cancel it out, get a general solution in terms of [tex]t[/tex], and plug in the boundary values to find my constants.