# 16bf10lec15 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon () Math 16b Autumn 2010 1 / 25 Linear di erential equations A rst order linear di erential equation is a di erential equation of the form y + a ( t ) y = b ( t ) Thomas Scanlon () Math 16b Autumn 2010 2 / 25 Example Solve the di erential equation y + ty = Thomas Scanlon () Math 16b Autumn 2010 3 / 25 Solution Subtracting ty from both sides, we have the equation y =- t y In this case we can use the method of separation of variables. If y is constant, then ty ≡ y ≡ 0 so that y ≡ 0. Thomas Scanlon () Math 16b Autumn 2010 4 / 25 Solution, continued Otherwise, we may express the equation as y y =- t Let C = y ( ) . Integrating with respect to t , we have- 1 2 T 2 = Z T- tdt = Z T y dt y = ln | y ( T ) C | (As our solution must be continuous and cannot take the value zero, the signs of y ( T ) and C = y ( ) must agree. So, we may drop the absolute value bars.) Exponentiating both sides of this equation and multiplying by C , we obtain y ( T ) = Ce- 1 2 T 2 . Thomas Scanlon () Math 16b Autumn 2010 5 / 25 Another Example Solve the di erential equation y + y = 10 e- t Thomas Scanlon () Math 16b Autumn 2010 6 / 25 Solution In this case, we cannot apply the separation of variables technique. However, as e t is never equal to zero, the solutions to the original equation y + y = 10 e- t and to the equation e t y + e t y = 10 are the same. Using the chain rule, we see that d dt ( e t y ) = e t y + e t y Thus, our new di erential equation is d dt ( e t y ) = 10 Thomas Scanlon () Math 16b Autumn 2010 7 / 25 Solution, continued From the equation d dt ( e t y ) = 10 we integrate with respect to t . e T y ( T )- y ( ) = e t y ( t ) | t = T t = = Z T d dt ( e t y ) dt = Z T 10 dt = 10 T So, if we write C = y ( ) , then we have y ( T ) = 10 e- T T + Ce- T ....
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## This note was uploaded on 02/08/2011 for the course MATH 16B taught by Professor Sarason during the Fall '06 term at University of California, Berkeley.

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16bf10lec15 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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