Discussion
The general solution of the differential equation \(\dfrac{d^{2}y}{dt^{2}} + \dfrac{dy}{dt} - 2y = 0\) is
Answer: C
The general solution is \(Ae^{-2t} + Be^{t}\).
Try \(y = e^{rt}\). The characteristic equation and its roots are:
\[ \begin{aligned} r^{2} + r - 2 &= 0 \\ (r + 2)(r - 1) &= 0 \\ r &= -2,\ 1 \end{aligned} \]
Two distinct real roots give \(y = Ae^{-2t} + Be^{t}\).
The other options use the wrong roots: \(Ae^{-t} + Be^{2t}\) solves \(r^{2} - r - 2 = 0\) (the sign of the middle term flipped), \(Ae^{-2t} + Be^{-t}\) needs roots -2 and -1, and \(Ae^{t} + Be^{2t}\) needs roots 1 and 2. A quick check is that the roots must add to -1 and multiply to -2.
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