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# lec16 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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± ² ± ² ± ² 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
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lec16 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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