Discussion
Consider the following statements :
(I) Every initial value problem can be reduced to a Fredholm integral equation.
(II) Every boundary value problem can be reduced to a Volterra integral equation.
Then :
Answer: D
Both (I) and (II) are wrong.
The pairing in the statements is reversed. An initial value problem, with all conditions at one point, integrates to a Volterra equation, whose upper limit is the variable \(x\). A boundary value problem, with conditions at both ends, leads to a Fredholm equation over the fixed interval \([a, b]\), using the Green's function.
For example, \(y'' + y = 0\), \(y(0) = 0\), \(y'(0) = 1\) becomes \(y(x) = x - \int_{0}^{x}(x - t)y(t)\,dt\), a Volterra equation. So neither statement is correct, and the options calling either correct are ruled out.
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