c The accumulation of lead is the simplest DE requiring only integration The DE

# C the accumulation of lead is the simplest de

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c. The accumulation of lead is the simplest DE, requiring only integration. The DE satisfies: dP dt = Kb k - q e - qt - e - kt , with P (0) = 0 . Upon integration we have P ( t ) = Kb k - q e - kt k - e - qt q ! + C, where the initial condition gives C = Kb k - q 1 q - 1 k . It follows that the solution satisfies: P ( t ) = Kb k - q e - kt k - e - qt q + 1 q - 1 k ! . e. With the linear ODE describing the weight of the boy is written: dw dt = - r ( w - w ) with w (0) = w 0 . Make the substitution z ( t ) = w ( t ) - w , where z (0) = w 0 - w . This creates the easier problem: dz dt = - rz with z (0) = w 0 - w .

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The solution is z ( t ) = w ( t ) - w = ( w 0 - w ) e - rt . It follows that w ( t ) = w + ( w 0 - w ) e - rt .
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