math 310 a3 Sol - MATH 310-3 Differential Equations...

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MATH 310-3 Fall 2011 Differential Equations Homework Set 3 Some Solutions The following solutions are not necessarily complete, but should highlight the main points for most questions. If you have any further questions regarding the problems and solutions, feel free to discuss them on WebCT. Solutions marked by * are taken from the publisher’s Instructor’s Solutions Manual for the textbook by Boyce and DiPrima, 9th edition (Wiley), while solutions marked by (which appear to have been typed with a typewriter. . . ) are from the publisher’s Student Solutions Manual; in some cases I have added comments in [brackets]. Section 2.1: Section 2.1 # 13 : Section 2.1 # 18 : [Hence the final solution is y ( t ) = 1 t 2 sin t - 1 t cos t + 1 t 2 ± π 2 4 - 1 ² ; when solving for y , remember to divide the constant by the integrating factor. . . ] * Section 2.1 # 24 :
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* Section 2.1 # 28 : [There is actually a quicker approach to finding y 0 that does not require differentiating the solution y ( t ): As discussed above, the solution touches but does not cross the t -axis at a value t = t 1 so that y ( t 1 ) = y 0 ( t 1 ) = 0. Now from the original differential equation y 0 + 2 3 y = 1 - t/ 2, we can see that y and y 0 can both be zero only if 1 - t/ 2 = 0, giving immediately t 1 = 2. Now substituting t = 2, y = 0 into the solution y ( t ) = (21 - 6 t ) / 8 + ( y 0 - 21 / 8) e - 2 t/ 3 derived above, we get 0 = 9 / 8 + ( y 0 - 21 / 8) e - 4 / 3 ; now we can solve for y 0 to get y 0 = 21 / 8 - (9 / 8) e 4 / 3 , as before.] Section 2.1 # 33 : Section 2.3: Section 2.3 # 4 : [ Q ( t ) is the quantity of salt in the tank, in pounds (lb), while the volume at time t (in minutes) is V ( t ) = 200 + 3 t - 2 t = 200 + t (in gallons, gal), and the concentration is consequently c ( t ) Q ( t ) /V ( t ) = Q ( t ) / (200 + t ) (in lb/gal). Solving the differential equation dQ dt + 2 200 + t Q = 3 , we use an integrating factor μ ( t ) = exp[ R 2 200+ t dt ] = exp[2 ln(200 + t )] = (200 + t ) 2 . Multiplying by the integrating factor, (200 + t ) 2 dQ dt + 2(200 + t ) Q = 3(200 + t ) 2 , or d dt
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This note was uploaded on 01/30/2012 for the course MATH 310 303 taught by Professor Rwk during the Spring '11 term at Simon Fraser.

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math 310 a3 Sol - MATH 310-3 Differential Equations...

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