Differential Equations midterm 1

sections 1.1 - 4.3

Exponential growth or decay

dP/dt = kP

P(t) = P₀e^kt

where k > 0 for growth, and k< 0 for decay

Newton’s Law of Cooling definition and equation

the rate of change in an objects temp is proportional to the difference between the temp of the object and the temp of its surroundings

dT/dt = k(T-T_s)

T(t) = T_s + (T₀ - T_s)e^k(t-t₀)

Mixture problems

dA/dt = rate in - rate out

rate in: (incoming solution concentration)(flow rate in)

rate out: (outgoing solution concentration)(flow rate out)

outgoing solution concentration = A(t)/V(t)

A(t) = amount of solution in tank at time t

V(t) = volume of mixture in tank at time t

to find V(t):

dV/dt = rate of mixture in - rate of mixture out

growth with carrying capacity

dP/dt = P(a-bP₀)

P(t) = (aP₀)/(bP₀ + (a-bP₀)e^-at)

a > 0 (birth rate)

b > 0 (death rate)

a/b = carrying capacity

Chemical reactions A + B = X

let:

A = amount of chemical A in reaction

aX = ratio of chemical A used in reaction

B = amount of chemical B in reaction

bX = ratio of chemical B used in reaction

dX/dt = k(A - aX)(B - bX) = kab(A/a - X)(B/b - X)

(A/a - X)/(B/b - X) = ce^kt

X(t) = (A/a - ce^kt(B/b))/(1 - ce^kt)