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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 pothole 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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 Fall '08
 BROWN
 Equations, Complex number

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