Discussion
The complete solution for the differential equation \(\dfrac{d^{2}y}{dx^{2}} + 2\dfrac{dy}{dx} - 3y = 0\) for \(y(0) = 0\) and \(\dfrac{dy}{dx}(0) = 8\) is __________.
Answer: B
The solution is \(y = 2(e^{x} - e^{-3x})\).
The auxiliary equation of \(y'' + 2y' - 3y = 0\) is
\[ m^{2} + 2m - 3 = (m - 1)(m + 3) = 0 \]
so \(m = 1\) or \(-3\) and \(y = Ae^{x} + Be^{-3x}\). Apply the conditions:
\[ \begin{aligned} y(0) &= A + B = 0 \\ y'(0) &= A - 3B = 8 \\ \Rightarrow A &= 2,\ B = -2 \end{aligned} \]
The options with \(e^{3x}\) take the wrong root (+3, from \(m^{2} - 2m - 3\)). \(2(e^{x} + e^{-3x})\) has the right roots but gives \(y(0) = 4\), not 0.
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