Solve first order difference equation

In summary, a first order difference equation is a mathematical equation that describes the relationship between consecutive values of a variable. Its general form is y<sub>n+1</sub> = f(y<sub>n</sub>) and it can be solved by finding the general solution and then using initial conditions to find the particular solution. This type of equation has many real-life applications, including in economics, biology, engineering, and physics. Some common techniques used to analyze it include finding the equilibrium point, stability analysis, and phase plane analysis.
  • #1
toofle
20
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Homework Statement


Solve the first order difference equation.

Homework Equations


x[n] - x[n-1] = n(n+1)/2
x[1] = 1



The Attempt at a Solution


Homogenous solution:
Characteristic equation:
r - 1 = 0 <=> r = 1
=> yhn = C * 1^n = C

Particular solution:
ypn = ?
I'm supposed to do some intelligent guess here. n(n+1)/2 is a polynomial of degree 2 and doesn't exist in yhn so i should be something like An(n+1) or An^2+Bn but Ican't get anything to work.
 
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  • #2
I would guess a polynomial of order 1 degree higher than n(n+1)/2. Say An^3+Bn^2+Cn.
 

FAQ: Solve first order difference equation

What is a first order difference equation?

A first order difference equation is a mathematical equation that describes the relationship between consecutive values of a variable. It is used to model situations where there is a change or difference between values over time or space.

What is the general form of a first order difference equation?

The general form of a first order difference equation is: yn+1 = f(yn), where yn+1 represents the value of the variable at the next time point and f(yn) is a function of the current value of the variable.

How do you solve a first order difference equation?

To solve a first order difference equation, you need to find the general solution by using techniques such as substitution, separation of variables, or using a specific formula for the given equation. Then, you can use initial conditions to find the particular solution.

What are some real-life applications of first order difference equations?

First order difference equations can be used in various fields such as economics, biology, engineering, and physics to model situations such as population growth, compound interest, chemical reactions, and electric circuits.

What are some common techniques used to analyze first order difference equations?

Some common techniques used to analyze first order difference equations include finding the equilibrium point, stability analysis, and phase plane analysis. These techniques help determine the behavior and stability of the solution to the equation.

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