Q2rs - F ( x,y ) = Z ( x 2 + 2 xy ) dx + h ( y ) = x 3 / 3...

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Quiz II 1. A 60 gallon tank contains 30 gallons of pure water at time t = 0. At that time a salt-water solution containing 0.25 lb/gal of salt begins to enter the tank at a rate of 4 gal/min. Simultaneously, a drain at the bottom of the tank is opened, allowing the salt-water solution to leave the tank at a rate of 2 gal/min. What is the salt content (in lbs) of the tank at the moment that it becomes full? Solution . This time the volume of the solution in the tank at time t (in minutes) is V ( t ) = 30 + (4 - 2) t gallons. So setting S ( t ) equal to the amount of salt in the tank at time t , you have dS dt = (0 . 25lbs/gal)(4 gal/min) - ( S ( t ) lbs V ( t ) gal )(2gal/min) . Rearranging and simplifying that gives you dS dt + S ( t ) 15 + t = 1 The integrating factor for that equation is (15 + t ) : (15 + t ) dS dt + S = 15 + t. (Note that (15 + t ) 0 = 1). So (15 + t ) S = 15 t + t 2 / 2 + C , and S (0) = 0 implies C = 0. The tank is full when t = 15 and S (15) = (225 + 225 / 2) / 30 = 11 . 25 lbs. (very salty water!)
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2 2. a) The differential form ω = ( x 2 +2 xy ) dx +( x 2 + y 2 ) dy is exact. Find a function F ( x,y ) such that dF = ω . Solution . Following the standard procedure
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Unformatted text preview: F ( x,y ) = Z ( x 2 + 2 xy ) dx + h ( y ) = x 3 / 3 + x 2 y + h ( y ) F y = x 2 + h ( y ) So we need h ( y ) = y 2 , which gives h ( y ) = y 3 / 3 + C and F ( x,y ) = x 3 / 3 + x 2 y + y 3 / 3 + C. b) The level curves of the function F ( x,y ) that you found in part a) are solutions to a rst order dierential equation. Write down that dierential equation. Solution . We have = 0 when y = y ( x ) is a solution to the dierential equation. So 0 = ( x 2 + 2 xy ) dx + ( x 2 + y 2 ) dy dx dx = [( x 2 + 2 xy ) + ( x 2 + y 2 ) dy dx ] dx So the dierential equation is 0 = ( x 2 + 2 xy ) + ( x 2 + y 2 ) dy dx or in normal form dy dx =-x 2 + 2 xy x 2 + y 2 . Of course, you could get that just by setting = 0, dividing by dx and then solving for dy dx . That dividing by dx step does not make sense, but the notation is designed so that it leads to the right answer!...
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Q2rs - F ( x,y ) = Z ( x 2 + 2 xy ) dx + h ( y ) = x 3 / 3...

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