# Solve the IVP: y 00 − 2y 0 + 17y = 0; y(π/4) = 1; y 0 (π/4) = −1

August 31, 2017

Question
Practice Midterm II

1. Solve the IVP: y − 2y + 17y = 0;

y(π/4) = 1;

y (π/4) = −1

2. Determine the homogenous O.D.E whose general solution is
y(x) = c1 ex + c2 e−x + c3 xe−x + e−x (A cos(x) + B sin(x))
3. A mass weighing 8 lb stretches a spring 2 ft. Assume there is no damping or external forces acting on
the system. Suppose the mass is pulled down 1 ft below its equilibrium position, and released with an
upward velocity of 4 ft/s.
(a) Determine the position y(t) of the mass at any time t.
(b) Find the amplitude, phase angle and period of the motion.
4. Consider the equation y − 2y + 2y = tet + et sin(t)
(a) Find the solution to the corresponding homogenous equation
(b) Using the Method of Undertermined Coeﬃcients (a.k.a the lucky guess), write down a particular
solution. Do NOT evaluate the coeﬃcients.
5. (a) Verify that y1 (t) = t solves the equation (for t > 0): t2 y + ty − y = 0. Find a second linearly
independent solution using Reduction of Order
(b) Let f (t) = t3 e3t . Find a particular solution to the equation t2 y + ty − y = f (t).
6. Find the general solution to the following homogenous problem: y (5) − 4y (4) + 4y (3) = 0
(Note that y (3) refers to the third derivative and so on…)
7. Using any method you like, ﬁnd a particular solution to y + 16y = cos4 (x)
(Hint: It is possible to guess a solution here but it is helpful to use some trig identities ﬁrst)

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