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20d_04s_2e - 3(9 points y = t and y = t 2 are solutions to...

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Math 20D Second Midterm 48 points May 26, 2004 Print Name, ID number and Section on your blue book. BOOKS and CALCULATORS are NOT allowed. One side of one page of NOTES is allowed. You must show your work to receive credit. Some integrals and derivatives that may be useful: integraltext tan x dx = - ln(cos x ) + C (tan x ) = sec 2 x (sec x ) = sec x tan x integraltext sec x dx = ln(sec x + tan x ) + C (arctan x ) = 1 1+ x 2 (arcsin x ) = 1 1 x 2 1. (27 points) Find the general solutions of the following differential equations. (a) y - (tan x ) y = 1 (b) y ′′ - 2 y + y = 4 t (c) ( x 2 + 1) y - y = 0 . 2. (3 points) I have decided to find a series solution y = a n x n to (4 + x 2 ) y ′′ + (1 - x ) y = 0 . For what values of x can you guarantee that the series will converge? Why? You must give a reason —the “Why?”—
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Unformatted text preview: 3. (9 points) y = t and y = t 2 are solutions to t 2 y ′′-2 ty ′ + 2 y = 0. Find a particular solution to t 2 y ′′-2 ty ′ + 2 y = t 2 e t . You may leave integrals in your answer. 4. (9 points) y ( t ) satis²es the di±erential equation y ′ ( t ) + y ( t-1) = 1 , with y ( t ) = 0 for t ≤ 0. Compute the Laplace transform Y ( s ) = L{ y ( t ) } . When your exam is returned, it will have the grade you will receive if you do NOT take the ²nal exam. In computing that grade, this exam will be weighted more than the ²rst exam to re³ect the fact that the ²nal will emphasize di±erential equations. END OF EXAM...
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