week05 - dy dx + p ( x ) y = q ( x ) multiply by the...

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Week 5: 2.5 Wronskian 3.1 Intro, Slope Fields, verify solution 3.2 Separable DE 3.4 Linear Equations ————– Problem: Verify that the function y = c 1 x is a solution of y ± = y 2 x Solution: Compute y ± and check. y ± = c 1 ( 1 2 ) x - 1 2 . y 2 x = c 1 x 2 x = c 1 ( 1 2 ) x ( x ) 2 = c 1 ( 1 2 ) 1 x = y ± . —————–
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2 Problem: Inflamable substance, temp T 0 = 50 (F), placed in hot oven, temp T m = 450 (F). After 20 min substance temp T = 150 . Find temp at 40 min. If substance ignites at 350, find time of combustion. Solution: Newton’s Law of cooling: T = T ( t ) temp at time t, dT dt = - k ( T - T m ) , T m = 450 , t in min. ————– dT dt = - k ( T - T m ) , T m = 450 , T (0) = 50 , T (20) = 150 , find T (40) and t c so T ( t c ) = 350 . Method: separation. dT T - T m = - k dt, ( T ± = T m ). integral: ln | T - T m | = - k t + c, T m = 450 . Initial Data: ln (450 - T ) = - k t + c, T (0) = 50 , T (20) = 150 . When t = 0 , ln 400 = c, e c = 400 . So 450 - T = e - kt e c = 400 e - kt , and T = T ( t ) = 450 - 400 ( e - k ) t . Next, when t = 20 , 150 = 450 - 400 ( e - k ) 20 , so - 300 = - 400 ( e - k ) 20 , e - k = ( 3 4 ) 1 20 , and T ( t ) = 450 - 400 ( 3 4 ) t 20 . Finally, T ( t ) = 450 - 400 ( 3 4 ) t 20 , gives T (40) = 225 , and T ( t c ) = 350 gives t c = 96 . 4 minutes. (why?) ————
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3 Main Step: to solve linear DE
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Unformatted text preview: dy dx + p ( x ) y = q ( x ) multiply by the integral factor f = e R p dx and use d dx ( fy ) = f ( y ± + py ) on the left to get d dx ( fy ) = fq. Problem: Solve dy dx + 2 x (1-x 2 ) y = 4 x,-1 ≤ x ≤ 1 . Solution: Find integral factor, inside integral first: Z 2 x (1-x 2 ) dx =-ln(1-x 2 ) = ln ( (1-x 2 )-1 ) (simplify!). ————– So e R 2 x ( 1-x 2 ) dx = e ln ( (1-x 2 )-1 ) = ( 1-x 2 )-1 = 1 1-x 2 = f. Multiply by f = 1 1-x 2 and use Main Property: 1 1-x 2 ± dy dx + 2 x (1-x 2 ) y ¶ = 4 x 1-x 2 , d dx ± y 1-x 2 ¶ = 4 x 1-x 2 . 4 Now integration gives y 1-x 2 = (-2 ln (1-x 2 )) + c, so y = (1-x 2 ) (-ln ((1-x 2 ) 2 ) + c ) . Notice that we can check this instance of the Main Property directly (using the product rule)....
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This note was uploaded on 02/29/2008 for the course MATH 205 taught by Professor Zhang during the Spring '08 term at Lehigh University .

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week05 - dy dx + p ( x ) y = q ( x ) multiply by the...

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