Discussion
For the integral equation
\[ x(t)=\frac{1}{2}\int_{t}^{1}(t\sqrt{s}-s\sqrt{t})\,x(s)\,ds \]
consider the following statements. The above integral equation is :
(I) Fredholm integral equation of second kind.
(II) Volterra integral equation of first kind.
Then :
Answer: D
Both (I) and (II) are false.
The upper limit is 1 but the lower limit is the variable t, so the range of integration depends on t: that makes it a Volterra-type equation, not a Fredholm equation (which has fixed limits). So (I) is false.
The unknown x appears outside the integral as well as inside it, which makes the equation of the second kind. A first-kind equation has the unknown only under the integral sign. So (II) is false as well; the equation is a Volterra equation of the second kind.
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