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111_pdfsam_math 54 differential equation solutions odd

# 111_pdfsam_math 54 differential equation solutions odd -...

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Exercises 3.2 find x ( t ). We can determine the concentration of salt in the first tank by dividing x ( t ) by the its volume, i.e., x ( t ) / 60 kg/gal. Note that the volume of brine in this tank remains constant because the ﬂow rate in is the same as the ﬂow rate out. Then output rate 1 = (3 gal / min) · x ( t ) 60 kg / gal = x ( t ) 20 kg / min . Since the incoming liquid is pure water, we conclude that input rate 1 = 0 . Therefore, x ( t ) satisfies the initial value problem dx dt = input rate 1 output rate 1 = x 20 , x (0) = x 0 . This equation is linear and separable. Solving and using the initial condition to evaluate the arbitrary constant, we find x ( t ) = x 0 e t/ 20 . Now, let y ( t ) denote the mass of salt in the second tank at time
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