$y = a e^{mx} + b e^{-mx}$ is the solution of the differential equation:
Options:
(A) $\frac{dy}{dx} - my = 0$
(B) $\frac{dy}{dx} + my = 0$
(C) $\frac{d^2y}{dx^2} + m^2 y = 0$
(D) $\frac{d^2y}{dx^2} - m^2 y = 0$
Official Answer Key:
D
Step-by-Step Solution & Explanation
Differentiate twice: dy/dx = am·e^(mx) − bm·e^(−mx); d²y/dx² = am²e^(mx)+bm²e^(−mx) = m²(ae^(mx)+be^(−mx)) = m²y. So d²y/dx² − m²y = 0. This matches the characteristic-equation logic too: roots ±m give the auxiliary equation r²−m²=0, i.e., d²y/dx²−m²y=0. Answer: (D). *Distractor note:* (C) has the sign of m² flipped, which would instead give oscillatory solutions y=A cos(mx)+B sin(mx), not exponentials.
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