practice_prelim2

# practice_prelim2 - Practice problems for Prelim 2 Math 293...

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Unformatted text preview: Practice problems for Prelim 2 Math 293 Fall 2005 1. Consider the difierential operator My) = (D — 1)2(D + 3)(y)- (a) Find the general solution to L(y) = 0. (b) Write down an appropriate yp(:1:), with undetermined coefﬁcients, which may be used to ﬁnd a solution to L(y) = seem—knew. Do not solve for the undetermined coefﬁcients. 2. Determine whether the following sets of functions are linearly dependent: l7 Points] (a) {32" lxl} [7 points} (b) {sin 9:, cos :17} points] (0) {1, e“, 6295} [20 points] 3. Find the general solution of the following ODE: y'” + 27y :2 0. {20 points} 4. Solve the initial value problem: y” +y = 1 + sinx, y(0) =2 O, y’(0) z 1. [20 points] 5. A Vibrating system satisﬁes the equation 3/” + cy' + y = 2, where 0 < c < 2. Find the value of the damping coefﬁcient for which the pseudoperiod of the damped oscillations is 50% greater than the period of the corresponding undamped motion. “N [4 points] [4 points] [4 points] [4 points] [4 points] [10 points] [10 points] [20 points] [20 points] 1. Circle one answer (TRUE or FALSE) for each of the following questions. (1) The equation (:c+y)dcc+:cdy=0 is an example of an exact ﬁrst order ODE. TRUE FALSE (2) The equation 23.; 2 Ma, used to model population growth, is called the logistic equation. TRUE FALSE (3) A second order linear homogeneous ODE always has two linearly independent so lutions. TRUE FALSE (4) If two functions f (:c), g(x) are linearly independent then their Wronskian W( f , g) is nonzero for all z. TRUE FALSE (5) Suppose that and f are periodic functions with F’(x) = f One may always obtain the Fourier series for by term—by~term integration of the Fourier series of ﬂat). TRUE FALSE 2. (a) Show that there exist at least two solutions to the initial value problem y(0) = O. (b) Explain why this does not contradict the theorem on existence and uniqueness for ﬁrst order ODE. 3. Solve the following initial value problem: 9331' + 6y = 333y4/3, y(1) = 2. 4. Show that the escape velocity from a spherical planet of mass M and radius R is given by /2GM R 7 where G is the universal gravitational constant. [HINTSz the distance r(t) of a projectile from the center of the planet satisﬁes the ODE r” = ~GM/T2. The escape velocity is the minimum velocity with which a projectile must be launched from the surface of the planet in order for its velocity v = r’ to remain positive for all time] [10 points] 5. (a) Find the general solution of y” + 23/ + 5y 2 0. [10 points] (b) Find the general solution of y” + 23/ + 5y = ex sings. [20 points] 6. Recall that the motion of a simple pendulum of length L is modelled by the differential equation L6” + 96’ = 0. Suppose a grandfather clock is gaining 15 minutes per hour (if it is set correctly at noon, then at 1 o’clock the grandfather clock says it is a quarter past one). Its pendulum length is 30 inches. How should the length be adjusted for the clock to keep perfect time? ...
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