I need a serious help please
Let Y be the metabolic rate of an organism, Yc the metabolic rate of a single cell, Nc(t) the total number of cells at time t, mc the mass of a cell, and Ec the energy required to create a new cell. The cell properties, Ec, mc, and Yc, are assumed to be constant and invariant with respect to the size of the organism. Thus
Y = YcNc + Ec(dNc/dt).
Let m be the total body mass of the organism at time t, and m = mcNc.(Note that Nc is the total number of cells in a body and is proportional to mass m, while the total number of capillaries Nn is proportional to 3/4 power of m.) Y=Y0(m)^(3/4)
a.show that the above equation can be written as
dm/dt = am^(3/4) - bm
with a = Y0mc / Ec and b = Yc/Ec
b.Let m = M be the mass of a matured organism, when it stops growing (dm/dt = 0). Find M, and show that the above equation can be rewritten as
dm/dt = am^(3/4)[1-(m/M)^(1/4)].
c.Let r = (m/M)^(1/4), and R = 1-r. Then the above equation becomes
dR/dt = -(a/4M^(1/4))R.
Solve this simple ordinary differential equation and show that a plot of ln(R(t)/R(0)) vs. the no-dimensinal time at/(4M^(1/4)) should yield a straight line with a slope -1 for any organism regardless of its size.
d. Based on this scaling for time t, argue that, for a mammal, the interval between heartbeats should scale with its size as M^(1/4).
I am completely lost,,,, please help ,,,, Thank you
Since every question is related to the above equation, finishing this should allow you to do the rest.