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GATE 2022 AE – Question 38

Engineering Mathematics · Differential Equations: Classification of PDEs, wave, Laplace and heat equations by separation of variables · 2 marks · Multiple choice

For $f_{xx}+f_{yy}=0$ and real separated solution $f=a(x)b(y)$, which statement can be true?

  1. a(x) is a periodic function and b(y) is a linear function
  2. both a(x) and b(y) are exponential functions
  3. a(x) is a periodic function and b(y) is an exponential function
  4. both a(x) and b(y) are periodic functions

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Show answer and explanation

Correct answer: (C) a(x) is a periodic function and b(y) is an exponential function

Explanation

**Separation of variables.** For $f=a(x)\,b(y)$ in Laplace's equation $f_{xx}+f_{yy}=0$:
$$a^{\prime\prime}b+ab^{\prime\prime}=0\;\Rightarrow\;\frac{a^{\prime\prime}}{a}=-\frac{b^{\prime\prime}}{b}=-\lambda .$$

**Case $\lambda>0$:** $a^{\prime\prime}=-\lambda a$ gives sinusoidal $a(x)$ (periodic), and $b^{\prime\prime}=\lambda b$ gives exponential (hyperbolic) $b(y)$.

**Case $\lambda<0$:** the roles are swapped: $a(x)$ is exponential and $b(y)$ is periodic.

**Case $\lambda=0$:** both are linear.

The two factors can never both be periodic or both exponential, since the second derivatives must have opposite signs. Of the options, the one that can be true is **a(x) periodic and b(y) exponential** (C). Option A (periodic and linear) is impossible, since $\lambda=0$ makes both linear.