Subtopic: Linear and non-linear first order Ordinary Differential Equations (ODE)
Question Statement:
The general solution of the differential equation $\frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 4y = 0$ is:
Options:
(A) $y = A e^{-x} + B e^{4x}$
(B) $y = A e^x + B e^{-4x}$
(C) $y = A e^{-x} + B e^{-4x}$
(D) $y = A e^x + B e^{4x}$
Official Answer Key:
A
Step-by-Step Solution & Explanation
Reading the equation as d²y/dx² − 3(dy/dx) − 4y = 0, the auxiliary (characteristic) equation is r² − 3r − 4 = 0, which factors as (r−4)(r+1) = 0, giving roots r = 4 and r = −1. For two distinct real roots, the general solution is y = Ae^(r₁x) + Be^(r₂x) = Ae⁻ˣ + Be⁴ˣ. Match this expression to whichever lettered option shows this exact exponent pattern in the original exam.
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