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Part CCSIR NET December 2024the-resolvent-exponentials-cancel-at-this-lambda

The resolvent exponentials cancel at this lambda

Let R(x, t) and u(x) denote the resolvent kernel and the solution, respectively, of the Volterra integral equation u(x) = eˣ +2∫(ln⁡6)xe(2(t−x))u(t)+ 2\int_(\ln 6)^x e^(2(t-x))u(t)dt. Then which of the following statements are true?

  1. A.R(x, t) = 1✓ Correct
  2. B.R(x, t) = e^(t−x)
  3. C.u(ln 8) = 10
  4. D.u(ln 7) = 9✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

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The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Volterra equations have eigenvalues just like Fredholm equations

More on this topic

The chapter behind this: Separable kernels and resolvent kernels — free to read

From Linear Integral Equations › Separable kernels and resolvent kernels

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