GATE 2026 ME – Question 37

Engineering Mathematics · Differential Equations: Heat, wave and Laplace equations · 2 marks · Multiple choice

Consider the following differential equation

$$\frac{\partial y}{\partial x} = 3\frac{\partial y}{\partial t} + y$$

If $y(x, 0) = 10e^{-2x}$, then the solution of the differential equation is

$y(x, t) = 10e^{-2x - t}$
$y(x, t) = 10e^{-2x + t}$
$y(x, t) = 10e^{-2x - 2t}$
$y(x, t) = 10e^{-2x + 2t}$

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Correct answer: (A) $y(x, t) = 10e^{-2x - t}$

Explanation

Try $y = 10e^{-2x + kt}$, which matches the initial condition at $t = 0$. Then $\frac{\partial y}{\partial x} = -2y$ and $\frac{\partial y}{\partial t} = ky$. The equation gives $-2 = 3k + 1$, so $k = -1$ and $y = 10e^{-2x - t}$.