General form of a Differential Equation Involving Growth and Decay
Growth and decay problems are commonly generalized under the exponential model,
would be the constant of proportionality.
Upon quick inspection, we can treat this model as a separable equation. Thus, the solution for this differential equation will be:
For IVPs, the solution would be,
where is the value/function of at a given time, and is a given value of time. We use because we solve for the value of at a given time period.
What is radioactive decay? Radioactive decay is a natural phenomenon of certain materials “losing” ( i.e. decaying) energy and matter over time due to their unstable atomic nucleus. Recall that atoms are made of particles called protons, neutrons, and electrons; under radioactive decay, these three particles are ejected out of the atomic nucleus, thus, a radioactive material will lose mass over time.
The general exponential decay function is defined as:
is the initial quantity, is the is treated as the “decay constant”, is the initial time (essentially zero in most cases), and would be any time duration.
Since , or the “half-life,” is the amount of time at which a radioactive material’s quantity is reduced to half, we can turn into,
where and .
We then solve for by canceling like terms and taking the natural logarithm of the equation:
((Recall that and ))
If a given problem indicates that lost a certain percentage, , of its mass at a certain time, then we can set .
For an IVP where and , the general solution is the same as the exponential decay function:
Carbon dating is a process based on the concept of radioactive decay. The unstable isotope, Carbon-14, decays into the stable isotope, Carbon-12. By determining the ratio of Carbon-14 and Carbon-12 in deceased organisms, scientists can determine the age of an organism– higher levels of Carbon-14 in a sample means that an organism died at a more recent time period.
Carbon dating uses the same function for radioactive decay problems, . Most problems, however, have us solve for . Also, unless otherwise stated, .
3.Continuously Compounding Interest
The concept of compounding interest applies to savings accounts, loans, credit cards, or most financial services that involves “interest.” Essentially, a certain amount grows in value at a specified percentage at specified intervals of time.
The general formula would be:
is the initial amount deposited or owed, is the annual rate or interest rate, is the number of times per year interest is compounded, and “t” would be the number of years.
Semi-annually would mean , quarterly is , monthly is , and daily is . Also note that should be written in it’s decimal form.
Therefore, if we infinitely compound interest, then we have this function,
The solution then to an IVP regarding continuously compounding interest where , , and is
4.Mixed Growth and Decay
Mixed growth and decay is a general term for problems that involve rate of changes that increase and decrease a value simultaneously.
The general forms would be:
rate of increase of – rate of decrease of
rate of increase of + rate of increase of
rate of decrease of – rate of decrease of
If one of the rate of changes are constant then, we’ll have
where would be a constant rate of change (negative or positive), and would be multiplied to a proportionality constant, which could also be negative or positive depending on the problem (recall the exponential model at the beginning of this post).
Notice that we now have a first order linear differential equation. To solve this type of problem, we can either use variation of parameters, separation of variables, integrating factors, transformations from Bernoulli equations and homogeneous equations, or substitution.