Question

Let \( u: \mathbb{R} \to \mathbb{R} \) be a twice continuously differentiable function such that \( u(0) > 0 \) and \( u'(0) > 0 \). Suppose \( u \) satisfies \[ u''(x) = \frac{u(x)}{1 + x^2} \]
for all \( x \in \mathbb{R} \).
Consider the following two statements:
I. The function \( uu' \) is monotonically increasing on \([0, \infty)\).
II. The function \( u \) is monotonically increasing on \([0, \infty)\).
Then which one of the following is correct?

• A. Both I and II are false.
• B. Both I and II are true.
• C. I is false, but II is true.
• D. I is true, but II is false.

Answer 1

The Correct Option is B

To determine which of the statements regarding the function \( u \) are true, we will examine both statements using the given differential equation:

The function \( u: \mathbb{R} \to \mathbb{R} \) is specified to be twice continuously differentiable, with \( u(0) > 0 \) and \( u'(0) > 0 \). It satisfies the differential equation:

\(u''(x) = \frac{u(x)}{1 + x^2}\) for all \( x \in \mathbb{R} \).

Statement I: The function \( uu' \) is monotonically increasing on \([0, \infty)\).

We need to evaluate the derivative of the function \( uu' \):

\(\frac{d}{dx}(uu') = u'u' + uu'' = (u')^2 + u\frac{u}{1+x^2}\).

This can be expressed as:

\(\frac{d}{dx}(uu') = (u')^2 + \frac{u^2}{1+x^2}\).

Since \( (u')^2 \geq 0 \) and \(\frac{u^2}{1+x^2} \geq 0\) for all \( x \), we have:

\(\frac{d}{dx}(uu') \geq 0\).

Thus, \( uu' \) is monotonically increasing on \([0, \infty)\).

Statement II: The function \( u \) is monotonically increasing on \([0, \infty)\).

To investigate, consider the sign of \( u'(x) \). Initially, \( u'(0) > 0 \) is given.

Additionally, from the differential equation \( u''(x) = \frac{u(x)}{1+x^2} \), and since \( u(x) > 0 \), it implies \( u''(x) \geq 0 \).

This indicates that \( u'(x) \) is non-decreasing, i.e., \( u'(x) \geq u'(0) > 0 \) for \( x \in [0, \infty)\).

Therefore, \( u \) is monotonically increasing on \([0, \infty)\).

Both statements I and II are verified to be true with the given conditions.

Therefore, the correct answer is:

Both I and II are true.