M427Kfin08 - y 00-xy 2 y = 0 about the x = 0 8 The function...

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M427K Final Exam Name NO NOTES. NO CALCULATORS. 1. Find the general solution of y 0 - 2 xy = x. 2. Use the method of undetermined coefficients to find the general solution of y 00 - y 0 - 2 y = 4 x 2 . 3. The function y = 1 is a solution to ty 00 - 2 y 0 = 0 . Find another solution and show it’s linearly independent to the first solution. 4. Let u ( x,t ) be a function of two variables. If possible, use the method of separation of variables to write u xx + u tt + 5 u = 0 as two different ordinary differential equations. 5. Use Laplace transforms to solve the IVP: y 0 + y = sin( x ) ,y (0) = 1 . 6. Find the eigenvalues and eigenfunctions of the BVP y 00 + λy = 0 ,y (0) = 0 ,y 0 (4) = 0 . 7. Find the recursion formula for the power series solution to
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Unformatted text preview: y 00-xy + 2 y = 0 about the x = 0 . 8. The function f ( x ) = 1 + (1 / 2) x on the interval 0 ≤ x ≤ 2 . Find the Fourier sine series of this function. 9. Let A = ± 2 1 0 2 ¶ . Solve x = A x , x (0) = ± 2 5 ¶ . 10. Determine whether x = 0 is a regular singular point of x 3 y 00 + 2 x 2 y + y = 0 . 11. (Ultramegabonus) Solve the following BVP involving the wave equation, where u ( x,t ) is the position of a string with fixed ends at time t . 4 u xx = u tt ,u (0 ,t ) = 0 = u (6 ,t ) ,u ( x, 0) = f ( x ), where f ( x ) = x on the interval 0 ≤ x ≤ 6....
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This note was uploaded on 05/02/2011 for the course M 427K taught by Professor Fonken during the Spring '08 term at University of Texas.

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