An annuity has a differential equation of y'=4500+.035y, and an initial condition of y(0)=1000. Describe what is going on.
To describe what is going on with the given annuity differential equation, let's break it down step by step.
The given differential equation is: y' = 4500 + 0.035y
In this equation:
- y' represents the rate of change of the annuity balance with respect to time. It signifies how the annuity balance is changing over time.
- '4500' represents a constant term, which means that the annuity balance is being increased by 4500 units regardless of its current value.
- '+ 0.035y' represents a term that depends on the current value of the annuity balance (y). It is proportional to the current balance with a proportionality constant of 0.035. This term accounts for the growth or decrease in the annuity balance based on its current value.
The given initial condition is: y(0) = 1000
This initial condition states that at time (t) equal to 0, the annuity balance (y) is equal to 1000 units. It provides a starting point for solving the differential equation.
By solving the differential equation with the given initial condition, we can find the specific solution that describes how the annuity balance changes over time. Various techniques, such as separation of variables, can be used to solve this type of differential equation.
To summarize, the given annuity differential equation and initial condition describe how the annuity balance (y) changes over time. The equation itself includes a constant term that adds a fixed amount to the balance and a term that accounts for the balance's growth or decrease based on its current value.