HW1 - into a first order separable variable equation, make...

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ME 391 : FALL 2007 HW 1 (Due Date: Sept. 12, 2007) 1. 0 2 2 = + + y y y a) Classify the above differential equation into the 4 different categories discussed in class. b) Show that both x be y x ae y x x sin , cos - - = = are solutions of the above equation. a and b are constants. c) Show that x be x ae y x x sin cos - - + = is also a solution. 2. The temperature distribution in a long solid cylinder with energy generation (which could represent a current-carrying wire or a fuel element in a nuclear reactor) is given by the differential equation k q dr dT r dr T d . 2 2 1 + + =0 , . q and k are constants The radius of the cylinder is r 0 . It is given that the temperature gradient at the centre of the cylinder is zero i.e. at r=0, 0 = dr dT and that the temperature at the surface is constant i.e. at r= r 0 , T= T s . Solve the differential equation to determine T as a function of r, utilizing the initial values given above. Hint: The second order differential equation cannot be integrated directly. In order to convert it
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Unformatted text preview: into a first order separable variable equation, make the substitution dr dT r T = and rewrite the equation in terms of T and r only. Solve it and using appropriate initial values, obtain a first order differential equation in T and r. Solve this equation for T(r) . 3. Solve the following equation for current i(t) in an electrical circuit with inductance L and resistance R. ) ( t v Ri dt di L = + The electromotive force is given by v(t)=V sin t and it is also known that at t=0, i=i . L,R and V are constants. 4. Show that the following equation is exact and solve it. dy xye x x dx e y y x x y xy xy ) 4 sin ( ) 2 cos sin 2 ( 2 2 2 2--= +-5. Practice Problems (do not submit for grading) Exercises 2.2: Problems 7,17,19,22,26,30 Exercises 2.3: Problems 2,6,12,22,28,30 Exercises 2.4: Problems 3,12,16,17,21...
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This note was uploaded on 07/25/2008 for the course ME 391 taught by Professor Blazer-adams during the Fall '08 term at Michigan State University.

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