Lecture 3 Notes

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Unformatted text preview: in the tank as a function of time. (b) What is the limiting mass of salt in the tank? (b) Directly from the equation (m’ = 400 - m/5), find an m for which m’=0. Modeling (Section 2.3) - Example • Saltwater with a concentration of 200 g/L flows into a tank at a rate 2 L/min. The tank starts with no salt in it and holds 10 L. The tank is well mixed and the mixed water drains out at the same rate as the inflow. (a) Write down an IVP for the mass of salt in the tank as a function of time. (b) What is the limiting mass of salt in the tank? (b) Directly from the equation (m’ = 400 - m/5), find an m for which m’=0. • m=2000. Called steady state - a constant solution. Modeling (Section 2.3) - Example • Saltwater with a concentration of 200 g/L flows into a tank at a rate 2 L/min. The tank starts with no salt in it and holds 10 L. The tank is well mixed and the mixed water drains out at the same rate as the inflow. (a) Write down an IVP for the mass of salt in the tank as a function of time. (b) What is the limiting mass of salt in the tank? (b) Directly from the equation (m’ = 400 - m/5), find an m for which m’=0. • m=2000. Called steady state - a constant solution. • What happens when m < 2000? ---> m’ > 0. Modeling (Section 2.3) - Example • Saltwater with a concentration of 200 g/L flows into a tank at a rate 2 L/min. The tank starts with no salt in it and holds 10 L. The tank is well mixed and the mixed water drains out at the same rate as the inflow. (a) Write down an IVP for the mass of salt in the tank as a function of time. (b) What is the limiting mass of salt in the tank? (b) Directly from the equation (m’ = 400 - m/5), find an m for which m’=0. • m=2000. Called steady state - a constant solution. • What happens when m < 2000? ---> m’ > 0. • What happens when m > 2000? ---> m’ < 0. Modeling (Section 2.3) - Example • Saltwater with a concentration of 200 g/L flows into a tank at a rate 2 L/min. The tank starts with no salt in it and holds 10 L. The tank is well mixed and the mixed water drains out at the same rate as the inflow. (a) Write down an IVP for the mass of salt in the tank as a function of time. (b) What is the limiting mass of salt in the tank? (b) Directly from the equation (m’ = 400 - m/5), find an m for which m’=0. • m=2000. Called steady state - a constant solution. • What happens when m < 2000? ---> m’ > 0. • What happens when m > 2000? ---> m’ < 0. • Limiting mass: 2000 g (Long way: solve the eq. and let t→∞.) Existence and uniqueness (Section 2.4) ∂f Theorem 2.4.2 Let the functions f and be continuous in some ∂y rectangle α < t < β , γ < y < δ containing the point (t0 , y0 ). Then, in some interval t0 − h < t0 < t0 + h contained in α < t < β, there is a unique solution y = φ(t) of the IVP y ￿ = f (t, y ), y (t0 ) = y0 . Existence and...
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This note was uploaded on 02/12/2014 for the course MATH 256 taught by Professor Ericcytrynbaum during the Spring '13 term at The University of British Columbia.

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