Chem Differential Eq HW Solutions Fall 2011 196

Chem Differential Eq HW Solutions Fall 2011 196 - A196...

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Unformatted text preview: A196 Appendix A Ordinary Differential Equations: Review of Concepts and Methods With the previous example in hand, we can solve Exercises 19-22 using Mathematica by repeating and modifying the commands. Here is an illustration with Exercise 19. We suppress some outcomes to save space. 19. y − y + 2y = ex , y(0) = 0, y (0) = 1. In[70]:= Clear y, seriessol, n, partsol n 10 seriessol Series y x , x, 0, n . y0 0, y ’ 0 leftside D seriessol, x, 2 D seriessol, x, 1 Series E ^ x, x, 0, n ; rightside equat LogicalExpand leftside rightside ; seriescoeff Solve equat ; ; partsol Normal seriessol . seriescoeff 1 Out[71]= x 2 seriessol; 10 Out[72]= 1 1 y 2 1 y 720 1 y 3 0 x3 6 y 7 0 x7 0 x6 5040 0 x2 6 1 1 y 4 0 x4 y5 24 120 y 8 0 x8 y 9 0 x9 40320 362880 0 x5 y 10 0 x10 3628800 Ox 11 The equation can be solved using analytical methods (undetermined coefficients). The exact solution is sol DSolve In[46]:= 1 14 sss x2 y ’’ x sol 7 Cos y’ x 2y x E ^ x, y 0 0, y ’ 0 1, 1, 2 7x 2 7 x2 Cos 7x 2 2 3 7 Sin 7x 2 Let's compare with the partial sum that we found earlier In[67]:= Plot sss, partsol , x, 2, 2 2 1.5 1 0.5 -2 Out[67]= 1 , y x ,x ; -1 Graphics We have a nice match on the interval [-2, 2] 1 2 7 x2 Sin 7x 2 2 ...
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