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GATE 2024 AE – Question 36

Engineering Mathematics · Differential Equations: First order linear ODEs and higher order linear ODEs with constant coefficients · 2 marks · Multiple choice

Given $y = e^{px}\sin qx$, where $p$ and $q$ are non-zero real numbers, the value of the differential expression

$\frac{d^2y}{dx^2} - 2p\frac{dy}{dx} + (p^2 + q^2)y$

is

  1. 0
  2. 1
  3. $p^2 + q^2$
  4. $pq$

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Show answer and explanation

Correct answer: (A) 0

Explanation

$y = e^{px}\sin qx$ is the imaginary part of $e^{(p + iq)x}$, which solves $m^2 - 2pm + (p^2 + q^2) = 0$ with $m = p \pm iq$. So $y$ is a solution of $y'' - 2py' + (p^2 + q^2)y = 0$, and the expression is 0. (Direct differentiation gives $y' = e^{px}(p\sin qx + q\cos qx)$ and $y'' = e^{px}[(p^2 - q^2)\sin qx + 2pq\cos qx]$, which also gives 0.)