# HW30 - | f x | is less than 10-4(b The function f x from...

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Assignment 1, due Monday Feb. 7 MAT 2384C, Winter 2011 Professor Jason Levy, University of Ottawa, Be sure to put your student number on your assignment! 1. (a) Use the method of undetermined coeﬃcients to ﬁnd the general solution to y 00 - 4 y 0 + 3 y = 13 cos(2 x ) . (b) Find the solution satisfying the initial conditions y (0) = 4 / 5 , y 0 (0) = 11 / 5. 2. Use the method of variation of parameters to ﬁnd the general solution to x 2 y 00 - 4 xy 0 + 6 y = x - 1 / 2 . Solve the following diﬀerential equations (the roots should be easy to ﬁnd): 3 . y (5) - 3 y (3) - 4 y 0 = 0 4 . x 4 y (4) + x 3 y (3) - 2 x 2 y 00 + 2 xy 0 = 0 5. Use Gaussian quadrature, with n = 3, to estimate the integral Z 4 0 e x dx. 6. (a) Use Newton’s method to estimate a root of f ( x ) = x 6 - 4 x 2 + 2 near x = 1. Keep four decimal place accuracy, and ﬁnish when the value of
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Unformatted text preview: | f ( x ) | is less than 10-4 . (b) The function f ( x ) from part (a) has a root between 0 and 1, and another root between 1 and 2, since f (0) and f (2) are positive while f (1) is negative. In part (a) we estimated one of these roots. Now estimate the other using the bisection rule, ending when | f ( x ) | < 1 / 3. 7. (a) Calculate the Lagrange polynomial p ( x ) that ﬁts the following data points: (0 . 92 , 3 . 45) , (1 . 34 , 5 . 78) , (2 . 06 , 4 . 15) . (b) Use your answer in (a) to interpolate a value of the function at x = 1 . 25. 1...
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