PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Continuity & Differentiability
Let \(G: \mathbb{R} \to \mathbb{R} \) be the function given by
\(G(x) = \frac{1}{2} \left( e^x - e^{-x} \right) \quad {for } x \in \mathbb{R}. \)
Let\( h: \mathbb{R} \to \mathbb{R} \) be defined by
\(h(x) = \int_0^\pi G(x \sin \theta) \, d\theta \quad {for } x \in \mathbb{R}. \)
Then which of the following is true?
1
x h''(x) + h'(x) + x h(x) = 0 for all \(x \in \mathbb{R} \) .
2
x h''(x) - h'(x) + x h(x) = 0 for all \( x \in \mathbb{R} \).
3
x h'''(x) + h'(x) - x h(x) = 0 for all \(x \in \mathbb{R} \) .
4
h''(x) - h'(x) - x h(x) = 0 for all \(x \in \mathbb{R} \) .