HW2 - y x 2 are both solutions of y 00 p x y 2 y = 0 Find y...

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MAE 294A / SIO 203A – Introduction to Applied Mathematics I – Fall 2009 Homework # 2 Assigned: October 8 2009 Due: October 15 2009, in class. Each question is worth 5 points. 1. Find the solution to y 0 = y 2 - e x y + e x , with y (0) = 1 - e R 1 0 e e t dt · 2. Find the solution to y 0 = 2 + ye xy 2 y - xe xy , y (0) = 1 . The solution cannot be expressed as an explicit function y(x) but as the solution to an implicit equation F [ x,y ] = 0 which you should derive. 3. Find a Green’s function representation of the solution to the inhomogeneous boundary value problem x 2 y 00 + xy 0 - y = f ( x ) , y (1) = y ( ) = 0 . 4. Find the solution to y 000 + y = δ ( x ) which vanishes at x = ±∞ . Calculate R -∞ y ( x ) dx and R -∞ xy ( x ) dx (you should find a way to evaluate these integrals without having to calculate much). 5. Suppose y ( x ) and [
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Unformatted text preview: y ( x )] 2 are both solutions of y 00 + p ( x ) y +2 y = 0. Find y ( x ) and the values of p ( x ) for which this is possible. 6. We will show that Z ∞-∞ δ ( x-a ) f ( x ) dx = f ( a ) . Let us assume that f has a Taylor series expansion about the point x = a . Consider two continuous sequences that approach the δ function: δ ± = 1 √ π± e-( x-a ) 2 /± δ ± = ± 1 / (2 ± ) | x-a | < ± | x-a | ≥ ± . In both cases, if you compute Z ∞-∞ δ ± ( x-a ) f ( x ) dx for ± 6 = 0 the result is not f ( a ) but instead f ( a ) with some corrections. Estimate the first nonvanishing correction for each form of δ ± ....
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This note was uploaded on 03/16/2010 for the course MAE 294b taught by Professor Young,w during the Fall '08 term at UCSD.

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