Needs to oscillate 5 times per second. Amplitude envelope increases 25% every second. Amplitude is 1 at t =0. The derivative at t = 0 is 54.6371. the sinusoid has to be a cos(2*pi*5*t) for the exponential exp(at), it becomes exp(a) = 0.25 for t = 1, 25% increase. Hence a = ln(0.25) = -1.38629 So for a growing exponential exp(+1.38629*t) Hence my expression becomes exp(+1.38629*t) * cos(2*pi*5*t). t = 0 is equal to one. but the derivative at t=0 is 1.38629.