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Difference equations are an essential part of engineering and business/economics applications. See Theory of Difference Equation (from University of Toronto).
One common example is the 'home mortgage' problem.
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The 'home mortgage' problem.
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In a home mortgage involving fixed interest rate of R%, loan amount of P, and loan period of M months, let us find out
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The monthly payment b
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The remaining loan amount after nth monthly payment x(n).
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Solution
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Set up Difference Equation
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Solve the Equation
After the nth monthly payment, remaining loan amount = x(n).
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Interest on the remaining loan amount for one month = x(n) * R/100 * 1/12
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Interest plus remaining loan = x(n) + x(n) * (R/100) *1/12 = x(n) * (1 + R/1200)
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Let a = (1 + R/1200)
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At the end of (n + 1)th month, a payment of b is made
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Hence, the remaining loan amount after (n + 1)th monthly payment is:
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x(n + 1) = x(n) * (1 + R/1200) - b
x(n + 1) = x(n) * a - b
x(n + 1) - a * x(n) = - b................................(1)
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(1) is called a first order Difference Equation
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DIFFERENCE EQUATIONS