16f10lec10-2 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Separation of variables The method of separation of variables applies to di erential equations of the form y = p ( t ) q ( y ) where p ( t ) and q ( x ) are functions of a single variable. Thomas Scanlon Math 16b Example Find the general solution to the di erential equation y = ty 2 Any constant solution to this equation would have 0 ty 2 so that y 0. Avoiding the constant solution, we may divide both sides of the equation by y 2 and then we solve must solve t = y y 2 . Thomas Scanlon Math 16b Solution, continued t = y y 2 Integrating, we have T 2 2 = Z T tdt = Z T y ( t ) dt y ( t ) 2 Substituting, u = y ( t ) we have Z T y y 2 dt = Z y ( T y ( ) u- 2 du =- 1 u | y ( T ) y ( ) = 1 y ( )- 1 y ( T ) Thomas Scanlon Math 16b Solution, completed Setting C := y ( ) we have T 2 2 = Z T tdt = Z T y y 2 dt = 1 C- 1 y ( T ) Performing some algebra, we compute y ( T ) = 2 C 2- CT 2 Thomas Scanlon Math 16b General procedure To solve the di erential equation...
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16f10lec10-2 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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