Chem Differential Eq HW Solutions Fall 2011 172

Chem Differential Eq HW Solutions Fall 2011 172 - A172...

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Unformatted text preview: A172 Appendix A Ordinary Differential Equations: Review of Concepts and Methods The Wronskian The Wronskian is a determinat, so we can compute it using the Det command. Here is an illustration. Clear y sol1 DSolve y ’’ x yx C 1 Cos x yx 0, y x , x C 2 Sin x Two solutions of the differential equation are obtained different values to the constants c1 and c2. For Clear y1, y2 y1 x_ c1 Cos x ; y2 x_ c2 Sin x ; Their Wronskian is w x_ Det c1 c2 Cos x y1 x , y2 x 2 c1 c2 Sin x , y1 ’ x , y2 ’ x 2 Let's simplify using the trig identity cos^2 x + sin^2 x =1 Simplify w x c1 c2 The Wronskian is nonzero if c1\= 0 and c2 \= 0. Let us try a different problem with a nonhomogeneous Clear y sol2 DSolve y ’’ x yx 1 yx C 1 Cos x 1, y x , x C 2 Sin x Two solutions of the differential equation are obtained different values to the constants c1 and c2. For Clear y1, y2 y1 x_ 1 Cos x ; y2 x_ 1 Sin x ; These solutions are clearly linearly independent (one is not a multiple of the other). Their Wronskian is Clear w w x_ Det Cos x y1 x , y2 x Cos x 2 Sin x , y1 ’ x , y2 ’ x Sin x 2 Let's simplify using the trig identity cos^2 x + sin^2 x =1 Simplify w x 1 Cos x Let's plot w(x): Sin x ...
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