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GATE 2016 CE (CE1) – Question 15

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

For $u_t=\alpha u_{xx}$, which is a possible separated solution?

  1. $C\cos(kt)[C_1e^{\sqrt{k/\alpha}x}+C_2e^{-\sqrt{k/\alpha}x}]$
  2. $Ce^{kt}[C_1e^{\sqrt{k/\alpha}x}+C_2e^{-\sqrt{k/\alpha}x}]$
  3. $Ce^{kt}[C_1\cos(\sqrt{k/\alpha}x)+C_2\sin(-\sqrt{k/\alpha}x)]$
  4. $C\sin(kt)[C_1\cos(\sqrt{k/\alpha}x)+C_2\sin(-\sqrt{k/\alpha}x)]$

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Correct answer: (B) $Ce^{kt}[C_1e^{\sqrt{k/\alpha}x}+C_2e^{-\sqrt{k/\alpha}x}]$

Explanation

Set u=X(x)T(t). Dividing ut=αuxx by αXT separates the variables, with T-prime/T=k and X-double-prime/X=k/α under the stated positive-k convention. Thus T is proportional to exp(kt) and X is a linear combination of exp(±sqrt(k/α)x), matching **B**. Boundary conditions would determine admissible constants and modes.