pset_6sol

pset_6sol - ACM 95/100b Problem Set 6 Solutions Faisal...

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ACM 95/100b Problem Set 6 Solutions Faisal Amlani February 26, 2009 Problem 1 (10 points) Consider the boundary value problem y ′′ + 2 y + y = x 0 x 1 y (0) = 0 y (1) = 0 The objective of this problem is to solve the BVP above using Greens functions. (a) Find the Greens function for this ODE. That is, solve the boundary value problem G ′′ + 2 G + G = δ ( x - x ) G (0 , x ) = 0 G (1 , x ) = 0 (b) Using the Greens function write down the formal solution to the problem y ′′ + 2 y + y = f ( x ) 0 x 1 Your answer should be in the form of an integral. (c) Finally, using the results above solve y ′′ + 2 y + y = x 0 x 1 SOLUTION ( a ) : The characteristic equation for the homogeneous case G ′′ + 2 G + G = 0 , assuming a solution of the form exp( αx ) , is clearly α 2 + 2 α + 1 = 0 = α = - 1 . Thus one homogeneous solution is exp( - x ) and by variation of parameters the other solution is x exp( - x ) . Thus for x < x , G ′′ + 2 G + G = 0 = G = a 1 exp( - x ) + a 2 x exp( - x ) . The only boundary condition applicable for x < x is G (0 , x ) = 0 which implies a 1 = 0 and hence a solution of the form G = c 1 x exp( - x ) . For x > x , G ′′ + 2 G + G = 0 = G = b 1 exp(1 - x ) + b 2 (1 - x ) exp(1 - x ) . The right boundary condition G (1 , x ) = 0 implies b 1 = 0 and hence a solution of the form G = c 2 (1 - x ) exp(1 - x ) . Thus we have that x 11 = c 1 x exp( - x ) , x 12 = c 2 (1 - x ) exp(1 - x ) .
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This note was uploaded on 12/12/2009 for the course ACM 95b taught by Professor Nilesa.pierce during the Winter '09 term at Caltech.

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pset_6sol - ACM 95/100b Problem Set 6 Solutions Faisal...

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