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Part BCSIR NET December 2023

Fredholm and Volterra equations

The value of $λ$ for which the integral equation $y (x) = λ \int_{0}^{1} x^{2}$ $e^{x + t}$ y(t) dt has a non-zero solution is

A. $4/ (1 + e^{2})$
B. $2/ (1 + e^{2})$
C. $4/ (e^{2} - 1)$ ✓
D. $2/ (e^{2} - 1)$

Solution

Separable kernel: $y = c x^{2} e$ ˣ with $c = λ c \int_{0}^{1} t^{2}$ $e^{2 t}$ dt $= λ c (e^{2} - 1) /4$ , so $λ = 4/ (e^{2} - 1)$ .

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

More on this topic

From Linear Integral Equations › Fredholm and Volterra equations

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