ps3solutions - (1+Ah) n = (1+At n /n) n is exp(At n ) ,...

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APMA 2101 Problem Set #3 Sample Solution Set (due 20 February 2007) Chapter 2: Review Problem For x(y) , we have dx/dy + x = y , with integrating factor exp(y) . Integrating both sides and using integration by parts once on the right-hand side gives x(y)=c*exp(-y)+y-1 . Section 2.4: Exact ODEs Section 6.1: The Euler Method 11. The exact solution, following the hint, is u(x) = tan(x+c) , or with the initial condition and a return to the original variables, y(x)=tan(x+ π /4)+1-x . Section 6.2: The Runge-Kutta Method
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Chapter 6: Supplementary Problems (a) The solution of y’=Ay with y(0)=y 0 is y(t)=y 0 exp(At) , so y(t n )=y 0 exp(At n ), where t n =nh. The limit of the binomial expansion
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Unformatted text preview: (1+Ah) n = (1+At n /n) n is exp(At n ) , which shows convergence to the analytic solution in the limit of infinitely fine stepping in the computational solution. (b) Asymptotically (which is where we always plan to be with numerical methods, for a sufficiently powerful computer on a fixed problem), we prefer Janes approach since squaring an error is much better than halving it after the error is already less than . Historical Research (a) Mechanica and The Theory of Motions of Rigid Bodies were two of Eulers books. (b) Euler claimed to be a holding a child in his hands while making some of his greatest discoveries....
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ps3solutions - (1+Ah) n = (1+At n /n) n is exp(At n ) ,...

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