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The 'home mortgage' problem: Boundary condition for (1)
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After M monthly payment remaining loan equals 0. Hence, x(M) = 0. This is the boundary condition.
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Also, x(0) = P, the principal loan amount, this is the initial condition.
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Solution of (1) using direct solution of Difference equation using shift operator E,
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Hence (1) becomes
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Homogenous part
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Auxilary equation
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Hence  where A is arbitrary constant.
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PARTICULAR SOLUTION
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Total solution
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 ..................................................................................(2)
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using
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in (2) we get
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 , remaining loan after nth monthly payment.....................(3)
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 , monthly payment...............................................(4)
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where P = principal loan amount, R =fixed interest rate %
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DIFFERENCE EQUATIONS