Discussion
The solution of the differential equation \(\frac{d^{2}y}{dt^{2}} - y\) = 0, subject to the boundary conditions y(0) = 1, \(y(\infty)\) = 0 is :
Answer: D
The solution is cosh t - sinh t.
The equation \(y'' - y = 0\) has the general solution \(y = Ae^{t} + Be^{-t}\). The condition \(y(\infty) = 0\) forces \(A = 0\), and \(y(0) = 1\) gives \(B = 1\), so \(y = e^{-t}\).
Since \(\cosh t - \sinh t = e^{-t}\), the answer is cosh t - sinh t. The option cosh t + sinh t equals \(e^{t}\), which grows without bound, and the trigonometric options solve \(y'' + y = 0\), a different equation.
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