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Diff Eq Exam 2 (2000) - .Section No April 4 2000 Name...

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Unformatted text preview: .Section No. April 4, 2000 Name: 110.302 Differential Equations Exam 2 No books, notes7 or calculators are permitted. (A short table of integrals and a Laplace transform table are provided.) Express all answers using real numbers only. (You may use complex numbers in your work, but 2' = \/—1 should not appear in your answer.) Each problem is worth 25 points. To receive full credit you must Show all of your work and write your answers in the provided spaces. Please put your name on the top of each page. Do not write here 1 1. Find the general solution of the differential equation // 2 I 2 y ——y +—2y=xcosx. x x (Give a precise solution; do not use infinite series.) Answer: y = Name: page 2 2. Fill in the boxes (with real numbers) in the following statement: The differential equation 92323;” + 9$2y' + 2y 2 0 has two Frobenius series solutions 211(56)::c2/3 (1+ 90+ 7 y2(x)=$1/3 56+ Show your work below: Name: page 3 3. a) Determine the Laplace transform Y(5) of the solution y(t) of the initial value problem y”—2y'+y=4et, y(0)=2,y’(0)=1. Answer: Y(s) = b) Use part (a) to find y. Answer: y(t) = Name: page 4 4. a) (10 points) Find the general solution of the differential equation y(4) + 25y = 0. Answer: y(t) = b) (15 points) A truck travelling on the Jones Falls Expressway goes over a pot-hole and starts to bounce up and down. Suppose that the truck’s suspension system exerts a vertical friction force with damping constant 7 = 200 kg/ sec and a vertical spring force with spring constant k; = 5kg/se02. With these fixed values of wk, the (quasi) frequency of the vertical oscillations depends only on the mass M of the truck. For what mass M are these oscillations the most rapid? (Consider only the vertical motion and ignore all other forces.) Answer: M = w kg. ...
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