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Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · ODE, PDE & Applied Mathematics › Linear Integral Equations

“Every integral equation of the second kind has a unique solution” — false

Counterexample: y(x) − 3∫₀¹ tx y(t)dt = eˣ

$λ = 3$ is an eigenvalue of the kernel tx, and eˣ is not orthogonal to t — no solution exists.

integral equations

#2 · ODE, PDE & Applied Mathematics › Linear Integral Equations

“Volterra equations have eigenvalues just like Fredholm equations” — false

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integral equations