A partial differential equation is given below. 2 u x2 - 2 u y2 = 0 Possible solution(s) is/are:

GATE 2026 CE (CE2) – Question 30

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

A partial differential equation is given below.

$$\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0$$

Possible solution(s) is/are:

$(x + y)^5$
$(x - 2y)^3$
$\cos(x + y)$
$\sin(x - 2y)$

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Show answer and explanation

Correct answer: (A) $(x + y)^5$; (C) $\cos(x + y)$

Explanation

This is the wave equation, and its general solution is $u = f(x + y) + g(x - y)$. Options A and C are functions of $x + y$, so they are solutions. For option B, $u_{xx} = 6(x - 2y)$ and $u_{yy} = 24(x - 2y)$, so $u_{xx} - u_{yy} \neq 0$. For option D, $u_{xx} = -\sin(x - 2y)$ and $u_{yy} = -4\sin(x - 2y)$, so again the difference is not zero.