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Unformatted text preview: 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 e as f ( s ) ds } for any constant c . Application to Solving for P (t) In The TwoState Chain In a twostate Markov chain the generator matrix is given by Q = parenleftbigg  parenrightbigg where and 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 ) = P 00 ( t ) + P 01 ( t ) = P 00 ( t ) + (1 P 00 ( t )) = ( + ) P 00 ( t ) + This is of the form 2. where x = P 00 ( t ), a = ( + ), and f ( t ) = . Thus the solution is....
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This note was uploaded on 08/01/2008 for the course STAT 333 taught by Professor Chisholm during the Spring '08 term at Waterloo.
 Spring '08
 Chisholm

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