Showing the temperature distribution in an infinitely long cylinder

[Rubik]

Homework Statement

Show that the temperature distribution in an infinitely long cylinder of metal with insulated sides and initial distribution

u(x, 0) = 0, the absolute value of x < L
u₀, the absolute value of x > L

where u₀ is a constant, is given for t >0, by

u(x, t) = (1/2)u₀[2 - erf((x +L)/(√4c²t)) + erf((x - L)/(√4c²t))

Homework Equations

u(x, t) = 1/√4∏c²t∫e^{-(x - y)2}/4c²t f(y) dy bounds = -∞<x<∞

The Attempt at a Solution

= (u₀/√4∏c²t)∫e^{-(x - y)24c2t}dy

Then I know you make a change of variables but I am having trouble with finding my new bounds of integration after I make the change of variables?

 

[Rubik]

relevant equation*

u(x, t) = (1/√4c²t)∫e^{-(x - y)2/4c2t}f(y) dy

Attempt at a solution*

= u₀/√4∏c²t)∫e^{-(x - y)2/4c2t}dy

Then I know I need to make a change of variables but after I do that I am not sure how to determine the new bounds of integration?