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# 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 differential equation with constant coefficients. (11b ) To be able to construct the solutions to such differential 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 differential equations with constant coefficients, and find 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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tut12 - The University of Sydney Math1003 Integral Calculus...

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