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lec16 - Lecture 16 18.01Fall 2006 Lecture 16 Dierential...

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Lecture 16 18.01Fall 2006 Lecture 16: Di ff erential Equations and Separation of Variables Ordinary Di ff erential Equations (ODEs) Example 1. dy = f ( x ) dx Solution: y = f ( x ) dx . We consider these types of equations as solved. Example 2. d + x y = 0 or dy + xy = 0 dx dx d ( + x is known in quantum mechanics as the annihilation operator. ) dx Besides integration, we have only one method of solving this so far, namely, substitution. Solving for dy gives: dx dy = - xy dx The key step is to separate variables . dy = - xdx y Note that all y -dependence is on the left and all x -dependence is on the right. Next, take the antiderivative of both sides: dy y = - xdx 2 x ln | y | = - 2 + c (only need one constant c ) | y | = e c e - x 2 / 2 (exponentiate) 2 y = ae - x / 2 ( a = ± e c ) c Despite the fact that e = 0 , a = 0 is possible along with all a = 0, depending on the initial conditions. For instance, if y (0) = 1, then y = e - x 2 / 2 . If y (0) = a , then y = ae - x 2 / 2 (See Fig. 1). 1
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Lecture 16 18.01Fall 2006 6 4 2 0 2 4 6 0 0.2 0.4 0.6 0.8 1 X Y 2 Figure 1: Graph of y = e - x 2 .
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