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Unformatted text preview: Lecture 16 18.01Fall 2006 Lecture 16: Differential Equations and Separation of Variables Ordinary Differential 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 = , a = 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)....
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.
- Fall '08