Integrating u''(y) \Phi(x-y) dy by parts | Math Homework Help

[squenshl]

Homework Statement

How do I integrate \int_a^b u''(y) \Phi(x-y) dy by parts.

Homework Equations

The Attempt at a Solution

I let \int_a^b u''(y) \Phi(x-y) dy = \int_a^x u''(y) \Phi(x-y) dy + \int_x^b u''(y) \Phi(x-y) dy but when I integrate by parts I get \int_a^b u''(y) \Phi(x-y) dy = \Phi(x-b)u'(b) - \Phi(x-a)u'(a) + \Phi'(x-b)u(b) - \Phi'(x-a)u(a) but I am missing out a u(x) term somewhere so I can write u(x) as a subject of everything else. Please help.

 

[SammyS]

What do you know about the function, \Phi(x)\ ?

 

[squenshl]

\Phi(x) = -|x|/2

 

[squenshl]

I think I got it, the u(y)\Phi&#039;(x-y) in the first integral becomes -u(x)/2 & the -u(y)\Phi&#039;(x-y) in the second integral becomes -u(x)/2 so adding these together we get -u(x) (assuming x < y) so u(x) can be written as a subject of the rest.