Let y = sin 2t and ze–t = 1 where t be the parameter. If \frac{d^{2} y}{\left(dz\right)^{2}} = – ae–bt (cos 2t + c sin 2t), a, b, c \in N then find (a + b + c).

Engineering

Mathematics

Successive Differentiation

Question

Let y = sin 2t and ze^–t = 1 where t be the parameter. If $\frac{d^{2} y}{{dz}^{2}}$ = – ae^–bt (cos 2t + c sin 2t), a, b, c $\in$ N then find (a + b + c).

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y = sin 2t, z = e^t
$\frac{{dy}_{t}}{{dz}_{t}} = \frac{2 \cos 2 t}{e^{t}}$ = 2 e^–t cos 2t
$\frac{d^{2} y}{{dz}^{2}} = \frac{d}{dt} (2 e^{- t} \cos 2 t) \times \frac{dt}{dz}$
$= \frac{(- 2 e^{- t} \cos 2 t - 4 e^{- t} \sin 2 t)}{e^{t}}$
= – 2(e^–2t cos 2t + 2e^–2t sin 2t)
a = 2, b = 2, c = 2
a + b + c = 6