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I. HODESIIVP with constant eoefi’icients. 1. Find a real valued solution to the following initial value problems. Sketch a graph of the solution. a. y” — 6y’ + 13y = 0, with y(0) = 1. HO) = 1- b. y” + 4y’ + 4y = 0, with y(0) = 1. y’CO) = -4- c. 6y” + 7y’ + 2y = 0, with 3’03) = 7- Y’Co) = ‘4- 2. For which values of a (if

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I. HODESIIVP with constant eoefi’icients.
1. Find a real valued solution to the following initial value problems. Sketch a graph of the solution. a. y” — 6y’ + 13y = 0, with y(0) = 1. HO) = 1-
b. y” + 4y’ + 4y = 0, with y(0) = 1. y’CO) = -4-
c. 6y” + 7y’ + 2y = 0, with 3’03) = 7- Y’Co) = ‘4- 2. For which values of a (if any) are all solutions of y” — (2a — 1)y’ + (:05: — 1) y = 0 unbounded as
t —> 00? 3. The characteristic equation of a homogeneous 9‘“ order linear Differential Equation with constant
coefficients has roots 1′ = 0 with multiplicity three, 1′ = —2 with multiplicity two, 1′ = —3 i 2i
with multiplicity two. Write the general solution of the Differential Equation. 4. One solution of the DE 6y”) + 531’” + 25y” + 2031′ + 4y 2 0 is y : cos(2x). Find the general
solution. 11. Reduction of order: 1. The ODE tzy" + Bty’ + y = 0 has a solution y1(t) = %for t > 0. Find the general solution. 2. The ODE Zty" — Sy’ + $31 = O has a solution y1(t) = t3 for t > 0. Find the general solution. III. Undetermined coefficients 1. Find the general solution ofthe ODE y” + Zy’ + y = 9—3. 2. Solve the WP: 31" —y’ — 2y 2 6x + 69”, 31(0) 2 1, y'(0) = 0.
3. Solve the WP: 31" —y’ — 2y = 6m“, y(0) = O, y’(0) = 1 4. Determine a suitable form for the particular solution Y(t) if the method of undetermined
coefficients is to be used. You do not need to determine the values of the coefficients. (i) y” + 3y’ 2 2152 + tie—35 + sin(3t) (ii) y" + y = t(1 + sin t) (iii) y” — Sy’ + 6y = e: cos(2t) + (3t + 4)e2t sin(t) (iv) 3/” + Zy’ + 231 = 3e“t + 29“t c050?) + 4.326% sin(t) (V) y" — 4y’ + 43,! = 2t2 + rite“ + tsin(2t) IV. Mass-Spring system 1. Consider the WP: y” + 4y = 0 with y(0) = —3 and y’(0) = 6. Write the solution as y(t) =
R cos(w0t — 6). 2. A mass of 2 kilograms stretches a spring 0.5 meters. If the mass is set in motion from its equilibrium
with a downward velocity of 10 cmls, and there is no damping, write an IVP for the position u (in
meters) of the mass at any time t (in seconds). Use g = 9.8 mfs2 for the acceleration due to gravity.

 
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