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Part CCSIR NET December 2023fredholm-alternative

Fredholm alternative

Consider the Fredholm integral equation y(x)−3∫01txy(t)y(x) - 3\int_{0}^{1} t x y(t) dt = f(x), with f continuous on [0, 1]. Which of the following choices for f(x) have the property that the equation admits at least one solution?

  1. A.f(x)=x2−1/2f(x) = x^{2} - 1/2✓ Correct
  2. B.f(x) = eˣ
  3. C.f(x) = 2 − 3x✓ Correct
  4. D.f(x) = x − 1

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 existence vs uniqueness.

See pricing

50 are analysed free — try those first.

The trap it tests

Existence vs uniqueness

A theorem giving one was read as giving both.

Drill statements like this

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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