GATE 2020 AE – Question 13

Engineering Mathematics · Differential Equations: First order linear ODEs and higher order linear ODEs with constant coefficients · 1 mark · Multiple choice

$y=Ae^{mx}+Be^{-mx}$, with constants A,B,m, solves which equation?

$y^{\prime\prime}-m^2y=0$
$Ay^{\prime\prime}+m^2y=0$
$By^{\prime\prime}+Ay=0$
$y^{\prime\prime}+my=m^2$

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Show answer and explanation

Correct answer: (A) $y^{\prime\prime}-m^2y=0$

Explanation

Differentiate twice: $y^{\prime\prime}=m^2Ae^{mx}+m^2Be^{-mx}=m^2y$. Rearranging gives the homogeneous ODE in A. The constants A and B weight independent solutions and do not belong as arbitrary coefficients of the governing equation.

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