Solving Initial Value Problem for Exponential Growth

In summary, the firm's sales department applies a continuous discount that decreases the unit price at a rate of $0.01 per unit ordered. The price per unit for a 100-unit order is $20.09, and the function p(x) can be calculated by solving the differential equation dp/dx = -p/100 with the initial condition p(100) = 20.09. The resulting equation is p = 20.09e^{1 - x/100}.
  • #1
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Homework Statement
To encourage buyers to place 100-unit orders, your firm's sales department applies a continuous discount that makes the unit price a function p(x) of the number of units x ordered. The discount decreases the price at a rate of $0.01 per unit ordered. The price per unit for a 100-unit order is p(100) = $20.09. Find p(x) by solving the following initial value problem:

Differential equation: dp/dx = -p/100
Initial condition: p(100) = 20.09


The attempt at a solution
So I integrated the differential equation and applied the initial condition to it. I was able to get the following equation:

[tex]p = 20.09e^{1 - x/100}[/tex]

Is this correct?
 
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  • #2
Yes, it's correct. You can verify it by confirming that it satisfies the differential equation and the initial condition.
 

FAQ: Solving Initial Value Problem for Exponential Growth

What is an initial value problem for exponential growth?

An initial value problem for exponential growth is a mathematical model that describes the behavior of a quantity that increases or decreases at a constant rate over time. The initial value, also known as the starting value, is the quantity at the beginning of the time period being studied.

How do you solve an initial value problem for exponential growth?

The first step in solving an initial value problem for exponential growth is to write the differential equation that represents the growth rate. Then, use techniques such as separation of variables or integration to find the general solution. Finally, use the given initial value to determine the specific solution for the problem.

What are some real-life applications of initial value problems for exponential growth?

Initial value problems for exponential growth are commonly used in fields such as finance, biology, and economics to model population growth, compound interest, and other exponential phenomena. They can also be applied to describe the spread of epidemics or the decay of radioactive materials.

How does the initial value affect the solution to an exponential growth problem?

The initial value is a crucial component of solving an initial value problem for exponential growth. It determines the starting point of the growth or decay and can significantly impact the behavior of the quantity over time. A small change in the initial value can result in a significant difference in the final solution.

Are there any limitations to using initial value problems for exponential growth?

While initial value problems for exponential growth can accurately model many real-world phenomena, they do have some limitations. For instance, they assume that the growth rate remains constant over time, which may not always be the case. Additionally, they may not be suitable for situations where the initial value is unknown or difficult to determine.

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