107_pdfsam_math 54 differential equation solutions odd

# 107_pdfsam_math 54 differential equation solutions odd -...

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CHAPTER 3:Mathematical Models and Numerical Methods Involving First Order Equations EXERCISES 3.2: Compartmental Analysis, page 98 1. Let x ( t ) denote the mass of salt in the tank at time t with t = 0 denoting the moment when the process started. Thus we have x (0) = 0 . 5 kg. We use the mathematical model described by equation (1) on page 90 of the text to Fnd x ( t ). Since the solution is entering the tank with rate 8 L/min and contains 0 . 05 kg/L of salt, input rate = 8 (L / min) · 0 . 05 (kg / L) = 0 . 4(kg / min) . We can determine the concentration of salt in the tank by dividing x ( t )bythevo lumeo
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Unformatted text preview: out. Therefore, the concentration of salt at time t is x ( t ) / 100 kg/L and output rate = x ( t ) 100 (kg / L) · 8 (L / min) = 2 x ( t ) 25 (kg / min) . Then the equation (1) yields dx dt = 0 . 4 − 2 x 25 ⇒ dx dt + 2 x 25 = 0 . 4 , x (0) = 0 . 5 . This equation is linear, has integrating factor µ ( t ) = exp ±R (2 / 25) dt ² = e 2 t/ 25 , and so d ( e 2 t/ 25 x ) dt = 0 . 4 e 2 t/ 25 ⇒ e 2 t/ 25 x = 0 . 4 ³ 25 2 ´ e 2 t/ 25 + C = 5 e 2 t/ 25 + C ⇒ x = 5 + Ce − 2 t/ 25 . Using the initial condition, we Fnd C . . 5 = x (0) = 5 + C ⇒ C = − 4 . 5 , 103...
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