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s08week06 - Week 6,7 2.5 Wronskian 3.1 Intro Slope Fields...

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Week 6,7: 2.5 Wronskian 3.1 Intro, Slope Fields, verify solution 3.2 Separable DE 3.3 Exact DE 3.4 Linear Equations 3.6 mixing/cooling ————– 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: Determine all values r so y = e rx is a solution to y - 4 y + 3 y = 0 . Main Step: to solve linear DE 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.
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3 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 . Now integration gives y 1 - x 2 = ( - 2 ln (1 - x 2 )) + c, so y = (1 - x 2 ) ( - ln ((1 - x 2 ) 2 ) + c ) .
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