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Differential

# Differential - STAT 333 Solutions To Differential Equations...

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STAT 333 Solutions To Differential Equations Suppose x = x ( t ) is a function of t . Then: 1. x prime = a x has general solution x = c e at where c is any constant. 2. x prime = a x + f ( t ) has general solution x = e at { c + integraltext t 0 e - as f ( s ) ds } for any constant c . Application to Solving for P (t) In The Two-State Chain In a two-state Markov chain the generator matrix is given by Q = parenleftbigg - λ λ μ - μ parenrightbigg where λ and μ can be any positive constants. The Kolmogorov Forward Equations are given by: P prime (t) = P (t) Q . We will solve these equations. Now since P 00 ( t ) + P 01 ( t ) = 1 and P 10 ( t ) + P 11 ( t ) = 1, there are only two unknowns P 00 ( t ) and P 11 ( t ). In pointwise form the Forward Equation for P prime 00 ( t ) looks like P prime 00 ( t ) = - λP 00 ( t ) + μP 01 ( t ) = - λP 00 ( t ) + μ (1 - P 00 ( t )) = - ( λ + μ ) P 00 ( t ) + μ This is of the form 2. where x = P 00 ( t ), a = - ( λ + μ ), and f ( t ) = μ . Thus the solution is given by P 00 ( t ) = e - ( λ + μ ) t { c + integraldisplay t 0 e ( λ + μ ) s μ ds } = e - ( λ + μ ) t { c + μ λ + μ e ( λ + μ ) t - μ λ + μ } = μ λ + μ + λ λ + μ e - ( λ + μ ) t where this last follows because P 00 (0) = 1 =

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Differential - STAT 333 Solutions To Differential Equations...

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