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Economics Dynamics Problems 98

Economics Dynamics Problems 98 - 82 Economic Dynamics 9 per...

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82 Economic Dynamics per gram. If this charcoal was formed during the reign of Hammurabi, what is the likely time of Hammurabi’s succession? 9. (i) f 1 ( x ) and f 2 ( x ) are linearly dependent if and only if there exists con- stants b 1 and b 2 not all zero such that b 1 f 1 ( x ) + b 2 f 2 ( x ) = 0 for every x . Suppose b 1 = 0, then f 1 ( x ) = − ( b 2 / b 1 ) f 2 ( x ) and so f 1 ( x ) is a multiple of f 2 ( x ) and therefore the functions are linearly dependent. If the set of functions is not linearly dependent, then f 1 ( x ) and f 2 ( x ) are linearly independent. Show that y 1 = e rt and y 2 = te rt are linearly independent. (ii) f 1 ( x ), f 2 ( x ) and f 3 ( x ) are linearly dependent if and only if there exist constants b 1 , b 2 and b 3 not all zero such that b 1 f 1 ( x ) + b 2 f 2 ( x ) + b 3 f 3 ( x ) = 0 for every x . If the set of functions is not linearly dependent, then f 1 ( x ), f 2 ( x ) and f 3 ( x ) are linearly independent. Show that y 1 = e rt , y 2 = te rt and y 3 = t 2 e rt are linearly independent. 10. For the second-order linear differential equation with complex conjugate roots r = α + β i and s = α β i , show that y ( t ) = c 1 e rt + c 2 e st is equivalent to y ( t ) = c 1 e α i cos( β t ) + c 2 e α i sin( β t ) by using Euler’s identity that for the complex number i β e i β = cos β + i sin
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