Explanation Given: The initial conditions are x ( 0 ) = x o , y ( 0 ) = 0 and z ( 0 ) = 0 . Procedure used: General differential equation: d y d x + P y = Q General solution equation: I y = ∫ I Q d t Radioactive decay of first element x: d x d t = − λ 1 x Radioactive decay of second element y: d y d t + λ 2 y = λ 1 x Radioactive decay of third element z: d z d t = λ 2 y Calculation: Write the expression for Radioactive decay of first element x . d x d t = − λ 1 x 1 x d x = − λ 1 d t (1) Integrate both sides of equation (1). ∫ 1 x d x = ∫ − λ 1 d t ln ( x ) = − λ 1 t + c 1 c 1 = ln ( x ) + λ 1 t c 1 = x e λ 1 t (2) Substitute boundary conditions t = 0 and x ( 0 ) = x o in equation (2). c 1 = x o e λ 1 × 0 c 1 = x o e 0 c 1 = x o Substitute x o for c 1 in equation (2). x o = x e λ 1 t x = x o e − λ 1 t Write the expression for Radioactive decay of second element y . d y d t + λ 2 y = λ 1 x (3) Substitute x o e − λ 1 t for x in equation (3). d y d t + λ 2 y = λ 1 x o e − λ 1 t Write the expression to calculate the integrating factor. I = e ∫ P d t (4) Compare equation (3) with general differential equation to obtain value of P as λ 2 and Q as λ 1 x o e − λ 1 t . Substitute λ 2 for P in equation (4). I = e ∫ λ 2 d t I = e λ 2 ∫ d t I = e λ 2 t Substitute e λ 2 t for I in general solution equation. e λ 2 t y = ∫ e λ 2 t λ 1 x o e − λ 1 t d t e λ 2 t y = λ 1 x o ∫ e ( λ 2 − λ 1 ) t d t e λ 2 t y = λ 1 x o ( λ 2 − λ 1 ) e ( λ 2 − λ 1 ) t + c 2 y = λ 1 x o ( λ 2 − λ 1 ) e ( − λ 1 ) t + c 2 e − λ 2 t (5) Substitute boundary condition y ( 0 ) = 0 in equation (5). 0 = λ 1 x o ( λ 2 − λ 1 ) e ( − λ 1 ) 0 + c 2 e − λ 2 0 0 = λ 1 x o ( λ 2 − λ 1 ) + c 2 c 2 = − λ 1 x o ( λ 2 − λ 1 ) Substitute − λ 1 x o ( λ 2 − λ 1 ) for c 2 in equation (5). y = λ 1 x o ( λ 2 − λ 1 ) e ( − λ 1 ) t − λ 1 x o ( λ 2 − λ 1 ) e − λ 2 t y = λ 1 x o ( λ 2 − λ 1 ) ( e ( − λ 1 ) t − e ( − λ 2 ) t ) Write the expression for Radioactive decay of third element z

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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