3.5 Method of Undetermined Coefficients - PS

3.5 Method of Undetermined Coefficients - PS - GE 207K...

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Unformatted text preview: GE 207K Method of Undetermined Coefficients PS October 03, 2011 Problem 1 Find the general solution to the following differential equation y 00- y = 1 . (1) Solution : The corresponding homogeneous differential equation is y 00- y = 0 , and the characteristic equation along with roots are r 2- r = 0 , r ( r- 1) = 0 , r 1 = 0 ,r 2 = 1 . Therefore, the homogeneous solution is y h = c 1 + c 2 e t . (2) For the given f ( t ) = 1 , we would choose y p = A . But a constant already appears in y h , therefore we apply the modification rule and multiply y p by t , hence obtaining ! y p = At. (3) Next we recheck to make sure our new y p doesnt have any terms which appear in y h . 3 The derivatives of y p are, y p = A, y 00 p = 0 . Plugging these into the ODE we find (0) |{z} y 00 p- A |{z} y p = 1 . A =- 1 . Therefore, the particular solution is y p =- t, (4) and the general solution is y = c 1 + c 2 e t- t . (5) fadaifard@utexas.edu 1 c hf, 2011 GE 207K Method of Undetermined Coefficients PS October 03, 2011 Problem 2 Find the general solution to the following differential equation y 00- 3 y- 4 y = 2 sin t (6) Solution : The corresponding homogeneous differential equation is y 00- 3 y- 4 y = 0 , and the characteristic equation along with roots are r 2- 3 r- 4 = 0 , ( r- 4)( r + 1) = 0 , r 1 = 4 ,r 2 =- 1 . Therefore, the homogeneous solution is y h = c 1 e 4 t + c 2 e- t . (7) For the given f ( t ) = 2 sin t , we choose y p = A cos t + B sin t. (8) By inspecting, we find that none of the terms in our choice for y p appear in y h . 3 The derivatives of y p are, y p =- A sin t + B cos t y 00 p =- A cos t- B sin t. Plugging these into the ODE we find (- A cos t- B sin t ) | {z } y 00 p- 3 (- A sin t + B cos t ) | {z } y p- 4 ( A cos t + B sin t ) | {z } y p = 2 sin t. Rearranging above equation by factoring our the functions, we have (- A- 3 B- 4 A ) cos( t ) + (- B + 3 A- 4 B ) sin t = 2 sin t + 0 cos t. (9) Comparing the coefficients of sin t and cos t on left and right sides of the equation, we note that- B + 3 A- 4 B = 2 ,- A- 3 B- 4 A = 0 . fadaifard@utexas.edu 2 c hf, 2011 GE 207K Method of Undetermined Coefficients PS October 03, 2011 Solving for A , and B we find that A = 13 17 , and B =- 5 17 . The particular solution is then y p = 13 17 cos t- 5 17 sin t, (10) and the general solution is y = y h + y p y = c 1 e 4 t + c 2 e- t + 13 17 cos t- 5 17 sin t . (11) fadaifard@utexas.edu 3 c hf, 2011 GE 207K Method of Undetermined Coefficients PS October 03, 2011 Problem 3 Find the general solution to the following differential equation y 00 + 4 y = 8 x 2 . (12) Solution : The corresponding homogeneous differential equation is y 00 + 4 y...
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3.5 Method of Undetermined Coefficients - PS - GE 207K...

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