The GATE Grind

GATE 2023 CE (CE1) – Question 25

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 1 mark · Multiple select

The following function is defined over the interval $[-L, L]$: $f(x) = px^4 + qx^5$.
If it is expressed as a Fourier series, $f(x) = a_0 + \sum_{n=1}^{\infty}\left\{a_n\sin\left(\frac{n\pi x}{L}\right) + b_n\cos\left(\frac{n\pi x}{L}\right)\right\}$, which options amongst the following are true?

  1. $a_n$, $n = 1, 2, \cdots, \infty$ depend on $p$
  2. $a_n$, $n = 1, 2, \cdots, \infty$ depend on $q$
  3. $b_n$, $n = 1, 2, \cdots, \infty$ depend on $p$
  4. $b_n$, $n = 1, 2, \cdots, \infty$ depend on $q$

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Correct answer: (B) $a_n$, $n = 1, 2, \cdots, \infty$ depend on $q$; (C) $b_n$, $n = 1, 2, \cdots, \infty$ depend on $p$

Explanation

The term $px^4$ is an even function, so it contributes only to the cosine coefficients $b_n$ (and $a_0$). The term $qx^5$ is odd, so it contributes only to the sine coefficients $a_n$. Hence $a_n$ depends on $q$ (B) and $b_n$ depends on $p$ (C).