# First Order Difference Equations

• diomedes.
In summary, Jack's wife Jill carried out 18 litres of water on the first day and the amount of water she carries out on the nth day is given by the difference equation An+1=0.96*An+2, with A1=18. To determine the total amount of water she recycles in the first week, one could manually solve for each day using the given equation or use the formula A(n)=Crn+D where C and D are constants. However, since this only involves a week, it is more efficient to simply use the difference equation for each day.
diomedes.

## Homework Statement

Background: Jack's wife Jill, mindful of the water restrictions is determined to carry buckets of bath water out of the house to water bean stalks.

She carries out 18 litres of water on the first day.

The amount of water Jill carries out An, in litres on the nth day is given by the difference equation: An+1=0.96*An+2 , A1=18

Specific: Determine the total amount of water Jill recycles in the first week.

## Homework Equations

An+1=0.96*An+2 , A1=18

## The Attempt at a Solution

Ok, well I've been working out the answer by finding the total amount recycled each day by manually putting the values into the formula: 1-18 ; 2-19.28 ; 3-20.51 ; 4-21.69 etc and then adding for the total (151.16), but this doesn't seem an efficient way of finding the answer. Is there a formula for the sum (similar to the sums of airthmetic/geometic sequences)?

It is quite possible the answer to this is impossible/obvious, so don't be too harsh :P :)

Have you learned how to solve difference equations? The solution is going to look like A(n)=Crn+D where C and D are constants, so it's the sum of a constant and a geometric progression.

Since $A_{n+1}= 0.96 A_n+ 2$ is a linear equation, we can add solutions to different parts. In particular, we can look for a solution to the equation $A_{n+1}= 0.96 a_n$ first. Since that involves just repeated multiplication, look for a solution of the form $A_n= A_1 r^n$ for some number r. Then $A_{n+1}= A_1r^{n+1}$ and the equation becomes $A_1r^{n+1}= 0.96 A_1r^n$. Solve that for r.

Now we could probably find a constant that satifies the entire equation, $A_{n+1}= .096 A_n+ 2$ by just substituting the constant, A, for both $A_{n+1}$ and $A_n$. Then add the two solutions.

But, frankly, since this only involves a week, just using the difference equation for each day is a perfectly valid way of solving this problem. You don't have to use a shotgun to kill flies!

Thanks for the advice chaps! Sounds like I'll just have to suck it up and solve for each day .

## 1. What is a first order difference equation?

A first order difference equation is a mathematical equation that describes the relationship between a sequence of values by using the difference between consecutive terms. It is used to model dynamic systems and predict their behavior over time.

## 2. What is the difference between a first order difference equation and a second order difference equation?

The difference between a first order difference equation and a second order difference equation is that a first order equation uses the difference between consecutive terms, while a second order equation uses the difference between consecutive terms and the difference between the previous term and the one before it.

## 3. How are first order difference equations used in real-world applications?

First order difference equations are used in various fields such as economics, biology, and physics to model the behavior of dynamic systems. They can be used to predict future values, analyze patterns, and make decisions based on the behavior of the system.

## 4. What are the key components of a first order difference equation?

The key components of a first order difference equation are the dependent variable, independent variable, and the difference operator (Δ) which represents the difference between consecutive terms. The equation may also include coefficients and initial conditions.

## 5. How are first order difference equations solved?

First order difference equations can be solved analytically using various techniques such as the method of undetermined coefficients or the method of generating functions. They can also be solved numerically using computational methods such as Euler's method or Runge-Kutta method.

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