2302-practice-final-soln

Cos x summationdisplay n 0 1 n x 2 n 2 n e x

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cos x = summationdisplay n =0 ( - 1) n x 2 n (2 n )! e x = summationdisplay n =0 x n n ! arcsinh x = summationdisplay n =0 ( - 1) n (2 n )! 4 n ( n !) 2 (2 n + 1) x 2 n +1 Variation of parameters ay ′′ + by + cy = f ( t ) has solution y ( t ) = C ( t ) y 1 ( t ) + D ( t ) y 2 ( t ) where C = integraldisplay - fy 2 a ( y 1 y 2 - y 2 y 1 ) dt D = integraldisplay fy 1 a ( y 1 y 2 - y 2 y 1 ) dt
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