varconcoeff - x ( t ) = Ke at + e at t Z e-a b ( ) d ....

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THE CASE OF FIRST ORDER LINEAR CONSTANT COEFFICIENTS: the initial value problem. THE EQUATION: dx dt = a x + b ( t ) THE INITIAL CONDITION : x (0) = x o THE HOMOGENEOUS PROBLEM: dx dt = a x . SOLUTION OF HOMOGENEOUS PROBLEM: x h ( t ) = K e at . DIFFERENTIAL EQUATION FOR K ( t ) : K 0 ( t ) = e - at b ( t ) . SOLUTION OF EQUATION FOR K : Any convenient solution will do, so we take the following one: K ( t ) = t Z 0 e - b ( τ ) 1
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THE GENERAL SOLUTION:
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Unformatted text preview: x ( t ) = Ke at + e at t Z e-a b ( ) d . SOLUTION OF THE INITIAL VALUE PROBLEM: Since x (0) = x o is the given initial condition and since x (0) = Ke + 0 = K , it follows that K = x o and the solution of the initial value problem is: x ( t ) = x o e at + e at t Z e-a b ( ) d . 2...
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This note was uploaded on 12/02/2009 for the course MATH 352 taught by Professor Staff during the Spring '08 term at University of Delaware.

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varconcoeff - x ( t ) = Ke at + e at t Z e-a b ( ) d ....

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