GATE 2024 CE (CE1) – Question 12

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

The second-order differential equation in an unknown function $u: u(x, y)$ is defined as $\frac{\partial^2u}{\partial x^2} = 2$. Assuming $g: g(x)$, $f: f(y)$, and $h: h(y)$, the general solution of the above differential equation is

$u = x^2 + f(y) + g(x)$
$u = x^2 + xf(y) + h(y)$
$u = x^2 + xf(y) + g(x)$
$u = x^2 + f(y) + yg(x)$

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Correct answer: (B) $u = x^2 + xf(y) + h(y)$

Explanation

Integrate once with respect to $x$: $\frac{\partial u}{\partial x} = 2x + f(y)$, where the "constant" can depend on $y$. Integrate again: $u = x^2 + xf(y) + h(y)$, with another arbitrary function of $y$.