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# tut10 - 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 10 2011 Assumed Knowledge Integration techniques. Objectives (9a) To be able to distinguish between separable and linear frst-order diFerential equations. (9b ) To be able to solve a linear equations utilising an integrating ±actor. Preparatory Questions 1. ²or each o± the ±ollowing equations, determine whether it is separable or a frst-order linear equation: ( i ) dy dx + 3 y = x ( ii ) t dx dt + x = cos t ( iii ) x 2 dy dx = x 2 - y ( iv ) dy dx = 2 x y 1 + x 2 . 2. Write each o± the linear equations in Question 1 in standard ±orm dy dx + p ( x ) y = q ( x ) (with a suitable renaming o± variables where necessary) and identi±y the ±unctions p and q . Practice Questions 3. ( i ) ²ind the general solution o± dy dx + 3 y = x . ( ii ) ²ind the general solution o± t dx dt + x = cos t ( iii ) ²ind the particular solution o± x 2 dy dx = x 2 - y ±or which y = 1 when x = 1. 4. ( i ) The direction feld ±or the equation dy dx = y - e - x is shown below. Sketch the graphs o± the solutions which satis±y the ±ollowing initial conditions: (a) x = 0, y = 0; (b) x = 0, y = 1.

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x y -3 3 -3 3 ( ii ) (a) Find the general solution to the above diferential equation.
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tut10 - The University of Sydney Math1003 Integral Calculus...

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