Volterra integral equation of the first kind

[James Briscoe]

Consider the integral equation:

/ x
| K(x,t)*theta(t)dt = f(x)
/ a

where the following are true:
a = -4
-4 < x < 0
K(x,t) = x + t
and
f(x) = 1

Two questions:
1) does this equation fit the definition of a
"Volterra integral equation of the first kind"?
2) what is theta(t)? (or can it be proven that given the above
conditions
theta(t) does not exist?)

[Julian V. Noble]

At the very least, f(a) = 0 is a necessary condition for consistency.

This equation is equivalent to [I call theta(x) = g(x) for ease of writing]

2xg' + 3g = f'' .

You get this by differentiating twice. I suggest you finish your HW
yourself by solving the above, substituting back into the Volterra
equation, and determining whether the result is consistent.

--
Julian V. Noble
Professor Emeritus of Physics
j...@lessspamformother.virginia.edu
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/

"For there was never yet philosopher that could endure the
toothache patiently."

-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.