The GATE Grind

GATE 2022 EC – Question 25

Engineering Mathematics · Differential Equations · 1 mark · Multiple select

Consider the following partial differential equation (PDE)

$$a\frac{\partial^2f(x,y)}{\partial x^2}+b\frac{\partial^2f(x,y)}{\partial y^2}=f(x,y),$$

where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that $f(x,y)=e^{(\xi x+\eta y)}$ is a solution of the given PDE.

  1. $\xi=\dfrac{1}{\sqrt{2a}},\ \eta=\dfrac{1}{\sqrt{2b}}$
  2. $\xi=\dfrac{1}{\sqrt a},\ \eta=0$
  3. $\xi=0,\ \eta=0$
  4. $\xi=\dfrac{1}{\sqrt a},\ \eta=\dfrac{1}{\sqrt b}$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\xi=\dfrac{1}{\sqrt{2a}},\ \eta=\dfrac{1}{\sqrt{2b}}$; (B) $\xi=\dfrac{1}{\sqrt a},\ \eta=0$

Explanation

Substituting $f=e^{\xi x+\eta y}$ gives $(a\xi^2+b\eta^2)f=f$, so we need $a\xi^2+b\eta^2=1$. (A): $\frac12+\frac12=1$ ✓. (B): $1+0=1$ ✓. (C): $0\ne1$. (D): $1+1=2\ne1$.