tut12 - The University of Sydney Math1003 Integral Calculus...

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The University of Sydney Math1003 Integral Calculus and Modelling Semester 2 Exercises for Week 12 2011 Assumed Knowledge Finding the roots of quadratic equations. Euler’s formula e = cos θ + i sin θ . Objectives (11a) To be able to write down the auxiliary (characteristic) equation associated with a second-order di±erential equation with constant coe²cients. (11b ) To be able to construct the solutions to such di±erential equations in terms of real exponential and trigonometric functions. Preparatory Questions 1. Write down the auxiliary (characteristic) equations for each the following second-order linear di±erential equations with constant coe²cients, and ³nd their roots: ( i ) d 2 y dt 2 + 2 dy dt - 8 y = 0 ( ii ) d 2 y dt 2 + 2 dy dt - 4 y = 0 ( iii ) d 2 y dt 2 - 9 y = 0 ( iv ) d 2 y dt 2 - 2 dy dt + 5 y = 0 ( v ) d 2 x dt 2 + 2 dx dt + x = 0 Practice Questions 2. Find the particular solution of Preparatory Question 1 ( i ) with y (0) = 0 and y (0) = 3.
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This note was uploaded on 09/01/2011 for the course YEAR 1 taught by Professor Various during the Three '11 term at University of Sydney.

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tut12 - The University of Sydney Math1003 Integral Calculus...

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