Question

Let H: $\R\rightarrow\R$ be the function given by H(x) = $\frac{1}{2}(e^x+e^{-x})$ for $x\isin\R$.
Let f: $\R\rightarrow\R$ be defined by
$f(x)=\displaystyle\int_{0}^{\pi}H(xsin\theta)d\theta$ for $x\isin\R$
Then which one of the following is true?

• A. xf"(x) + f'(x) +xf(x) = 0 for all $x\isin\R$.
• B. xf"(x) + f'(x) +xf(x) = 0 for all $x\isin\R$.
• C. xf"(x) + f'(x) +xf(x) = 0 for all $x\isin\R$.
• D. xf"(x) + f'(x) +xf(x) = 0 for all $x\isin\R$.

Answer 1

Answer: (C)

xf"(x) + f'(x) +xf(x) = 0 for all $x\isin\R$.

Answer 2

The correct option is (C): xf"(x) + f'(x) +xf(x) = 0 for all $x\isin\R$.