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Part CCSIR NET December 2025a-constant-kernel-has-a-constant-resolvent-it-cannot-take-different-values-at-different-points

A constant kernel has a constant resolvent it cannot take different values at different points

Let y(x) be the solution of the integral equation (IE) y(x) = eˣ +e+1+(1/3)∫01y(t)+ e + 1 + (1/3)\int_{0}^{1}y(t)dt, and R(x,t,1/3) be the resolvent kernel associated to IE. Then which of the following statements are true?

  1. A.R(0,1,1/3) = 3/2✓
  2. B.R(1/2,1/2,1/3) = 2/3
  3. C.y(1) = 3e+1✓
  4. D.y(1) = e+3

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.

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50 are analysed free — try those first.

Related counterexample: Every integral equation of the second kind has a unique solution

More on this topic

The chapter behind this: Integral equations: Fredholm vs Volterra — free to read

From Linear Integral Equations › Fredholm and Volterra equations

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