GATE 2020 PH – Question 11

Mathematical Physics · Linear differential equations: simple applications of first and second order linear differential equations and solutions · 1 mark · Multiple choice

Which function solves $d^2u/dx^2=k^2u$ for real k?

$e^{-kx}$
$\sin kx$
$\cos kx$
$\sinh x$

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Show answer and explanation

Correct answer: (A) $e^{-kx}$

Explanation

Two derivatives of $e^{-kx}$ give $k^2e^{-kx}$, satisfying the equation for every real k. Sine and cosine instead give $-k^2u$. The unscaled function $\sinh x$ solves $u\prime\prime=u$, and only satisfies the specified equation for special k. Thus A is the general valid option.

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