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Di¤erential Equations and Matrix Algebra I (MA 221), Fall Quarter, 1999-2000 WorkSheet 10 I) Solve the following di¤erential equations by hand. (a) dy x = dx y (b) y dy = ; dt t (c) dx = 2x dt (d) dy = 3y; dt (e) dy = ¡2y + 6 dx (f) dx = ¡2x + 6 ; dt y(1) = 1 y(0) = 4 x(0) = 9 II) Set us a di¤erential equation for each of the following situations. You need not solve the DE at this time. (a) Assume that the rate of growth of a bacteria is proportional to the amount present. If P (0) = 3g and P (1) = 3:2g; then …nd the amount at any time t: (b) Suppose that you deposit $2000 into an account that pays 7.5% interest compounded continuously. How much will be in the account 3 years later? (c) Assume that a cup of hot water (200 F) is put into a room (78 F) at noon on Friday. If the temperature at 12:15 is 120 F, …nd the temperature at any time t: Use Newton’s Law of Cooling. (d) A 100 gallon tank initially has a 100 gallon brine solution containing 20 lb of salt. If pure water enters the tank at a rate of 2 gal/min and the mixture leaves the tank at the same rate, …nd the amount of salt in the tank at any time t: (e) Same as (d) except assume that the water entering the tank contains 1/4 lb per gallon. Now …nd the amount of salt in the tank at any time t:

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