Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Linear Integral Equations Fredholm and Volterra Integral Equation
For the unknown y : [0, 1] → ℝ, consider the following two-point boundary value problem:
\(\rm \left\{\begin{aligned}\rm y^{\prime \prime}(x)+2 y(x) & =0 \quad \text { for } \rm x ∈(0,1), \\ \rm y(0) & =\rm y(1)=0 .\end{aligned}\right.\).
It is given that the above boundary value problem corresponds to the following integral equation:
y(x) = 2\(\displaystyle\int_0^1\) K(x, t) y(t) dt for x ∈ [0, 1].
Which of the following is the kernel K(x, t)?
1
K(x, t) = \(\begin{cases} \rm t(1-x) & \text { for } \rm tx\end{cases}\)
2
K(x, t) = \(\begin{cases} \rm t^2(1-x) & \text { for } \rm tx\end{cases}\)
3
K(x, t) = \(\begin{cases}\rm \sqrt{t}(1-x) & \text { for } \rm tx\end{cases}\)
4
K(x, t) = \(\begin{cases} \rm \sqrt{t^3}(1-x) & \text { for } \rm tx\end{cases}\)