hw0 - w t = 0 for all t ≥ 0 c[Uniqueness Say the...

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Math 425, Spring 2011 Jerry L. Kazdan Problem Set 0 [Rust Remover] DUE: Never. 1. Let u ( t ) be the amount of a radioactive element at time t and say initially, u ( 0 ) = A . The rate of decay is proportional to the amount present, so du dt = cu ( t ) , where the constant c determines the decay rate. The half-life T is the amount of time for half of the element to decay, so u ( T ) = 1 2 u ( 0 ) . Find c in terms of T and obtain a formula for u ( t ) in terms of T . 2. Let Z x 0 f ( t ) dt = e cos ( 3 x + 1 ) + A , where f is some continuous function. Find f and the constant A . 3. Say w ( t ) satisﬁes the differential equation aw 00 ( t )+ bw 0 + cw ( t ) = 0 , (1) where a and c , are positive constants and b 0. Let E ( t ) = 1 2 [ aw 0 2 + cw 2 ] . a) Without solving the differential equation, show that E 0 ( t ) 0. b) Use this to show that If you also know that w ( 0 ) = 0 and w 0 ( 0 ) = 0, then
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Unformatted text preview: w ( t ) = 0 for all t ≥ 0. c) [Uniqueness] Say the functions u ( t ) and v ( t ) both satisfy the same equation (1) and also u ( ) = v ( ) and u ( ) = v ( ) . Show that u ( t ) = v ( t ) for all t ≥ 0. 4. Say u ( x , y ) has the property that ∂ u ∂ y = 0 for all points ( x , y ) and that u ( x , ) = sin3 x . Find u ( x , y ) . What if instead u satisﬁes ∂ u ∂ y = 2 xy ? 5. A function u ( x , y ) satisﬁes u x + 3 u y =0. Find a change of variables x = as + bt y = cs + dt so that in the new ( s , t ) variables u satisﬁes ∂ u ∂ s = 0. [Last revised: January 15, 2011] 1...
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This document was uploaded on 03/06/2012.

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