1.49: (a) Show that Given that where and are integers (b) find the value of and the value of

4PM1/2/June/2025 — Question 9 · 8 marks

y=e−4tcos2ty = \mathrm{e}^{-4t} \mathrm{cos} 2t
(a) Show that 2e−4tsin2t=−dydt−4y\displaystyle 2\mathrm{e}^{-4t} \mathrm{sin} 2t = -\frac{\mathrm{d}y}{\mathrm{d}t} - 4y
(3)
Given that d2ydt2+Mdydt+Ny=0\displaystyle \frac{\mathrm{d}^{2}y}{\mathrm{d}t^2} + M \frac{\mathrm{d}y}{\mathrm{d}t} + Ny = 0 where M\displaystyle M and N\displaystyle N are integers
(b) find the value of M\displaystyle M and the value of N\displaystyle N
(5)