These are practice problems for the final exam. You should attempt all of them, but turn in only the even-numbered problems!
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1 Math 33 - ODE Due: 7 December 208 Written Problem Set # 4 Thee are practice problem for the final exam. You hould attempt all of them, but turn in only the even-numbered problem! Exercie Solve the initial value problem and ketch a graph of the olution.. 2y 3y + y = 0, y(0 = 2, y (0 = y y 2y = 0, y(0 =, y (0 = 2 3. y + 5y + 6y = 0, y(0 =, y (0 = 0 Exercie 2 Conider the initial value problem y + y 2y = 0, y(0 = 2, y (0 = β.. For which value of β the olution atifie lim t y(t = 0? 2. For which value of β doe the olution never hit 0? Exercie 3 An object tretche a pring 6 inche in equilibrium.. Set up the equation of motion and find it general olution. 2. Find the diplacement of the object for t > 0 if it i initially tretched upward 8 inche above equilibrium and given a upward velocity of 3 ft/. 3. Write down the olution found in 2. in the form R co(ω 0 t φ and determine the frequency, period, amplitude, and phae angle of the motion. Exercie 4 A 96 lb weight tretche a pring 3.2 ft in equilibrium. It i ubmitted to friction with damping contant c=8 lb-ec/ft. The weight i initially diplaced 5 inche below equilibrium and given a downward velocity of 2 ft/ec. Find it diplacement for t > 0. Exercie 5 Conider the pring ma ytem 4 d2 y 2 + k dy + 5y = 0, where k i a parameter with 0 k <. A k varie decribe the different type of the ytem (damped, overdamped, undamped. Determine for which k a bifurcation occur (thi mean that the ytem change it type at that value... Exercie 6 Conider the pring ma 2 + k dy + 2ky = 0, where k i a parameter with 0 k <. A k varie decribe the different type of the ytem (damped, overdamped, undamped. Determine for which k a bifurcation occur (thi mean that the ytem change it type at that value...
2 Exercie 7 Compute the general olution for the 2 + dy 6y = e 4t 2 + dy 6y = e2t 2 4dy + 5y = et + e 2t 2 4dy + 5y = 5 co(3t t2 Exercie 8 Conider the forced pring ma ytem where α i a parameter. + 5y = 6 in(αt. 2. For which value of α doe the ytem exhibit a reonance? 2. Find the general olution for the value of α found in Exercie 9 Conider the forced pring ma ytem. Find the general olution dy + 7y = 6 in(3t. 2. Decribe the behavior of the general olution a t (that i determine the teady-tate olution and graph a typical olution. 3. Compute the amplitude and phae angle of the teady tate olution. Exercie 0 Solve the initial value problem Exercie Conider the equation 2 4dy 5y = 6 in(3t, y(0 = 2, y (0 =.. Determine the frequency of the beating. + 9y = 6 in(3.t Determine the frequency of the rapid ocillation. 3. Give a rough ketch of typical olution indicating clearly the reult obtained in. and 2. Remark: To anwer thi quetion you do not need to compute the olution explicitly. Exercie 2 Find the general olution of dy2 + 6y = 3 in(4t. 2 dy2 + 6y = 5 co(2t 2 2
3 Exercie 3 Solve the initial value problem dy y = 3 in(4t, y(0 =, y (0 = 0 dy y = 5 co(2t, y(0 = 2, y (0 = 2 Exercie 4 Conider the linear ytem dy = AY where Y = ( y ( 2 In each cae of the five cae ( ( /4 7 2 y 2 and A i given by ( 2 0. Determine the type of the ytem, i.e., ink (node, ource, addle, center, piral ource, piral ink, center. 2. Draw the phae portrait of the ytem. If the eigenvalue are real you need to compute the eigenvector and indicate them clearly on the phae portrait. 3. Draw a rough graph of a typical olution y (t, y 2 (t. Note that you do not need to olve the ytem to do thi! If the eigenvalue are complex indicate clearly in your graph the period of the ocillation. Exercie 5 Conider the linear ytem dy ( ( = AY where A i given by ( /4 7 2 ( 2 0 In each cae. Compute the general olution of dy = AY. 2. Solve the initial value problem dy = AY, Y(0 = ( 2. Exercie 6 Conider the linear ytem dy ( α 2 2 α = AY where A i given by ( 4 α 2 and α i a parameter. Compute the eigenvalue a a function of α to determine the variou type of the ytem and the bifurcation a the parameter α varie. Exercie 7 Compute the invere Laplace tranform of the following function e 8 ( + 2 e /4 (e e 2 (g 2 5 (h e 3 ( 2 + ( e 2 (i ( ( e 5 (j ( (
4 Exercie 8 The function f(t i given by f(t = 0 if 0 t < 2 if t < 3 if 3 t.. Compute the Laplace tranform of f(t. Hint: Write f a a combination of u(t a for uitable a. 2. Solve the equation d2 y + 4y = f(t. 2 Exercie 9 The function f(t i given by f(t = { t if 0 t < 0 if t. Compute the Laplace tranform of f(t. Hint: Write f a a combination of u(t a for uitable a. 2. Solve the equation d2 y + 4y = f(t. 2 Exercie 20 Ue the Laplace tranform method to olve the following initial value problem dy + 5y = 5u(t 2, y(0 = 7. Make alo a graph of the olution. dy + 4y = 3u(t 4e2(t 4, y(0 = 2. What i lim t y(t? + 4y = 3u(t e (t y(0 2 large t? 2 + 2dy + 0y = u(t 4 y(0 of the olution. + 5y = δ(t 3 2 y( dy + 7y = δ(t 5 y(0 = 0, y (0 =. How doe the olution behave for = 2, y (0 = 0. What i lim t y(t? Make a graph = 2, y (0 =. Make a graph of the olution. = 6, y (0 =. Make a graph of the olution. 4
5 Table of Laplace Tranform f(t L(f(t f(t L(f(t t 2 Derivative t y L(y t n n! n+ y L(y y(o e at a y 2 L(y y(o y (0 t n e at n! ( a n+ co(ωt in(ωt coh(at inh(at e at co(ωt e at in(ωt 2 + ω 2 ω 2 + ω 2 t-shift 2 a 2 f(t F ( a 2 a 2 u a (tf(t a e a F ( a ( a 2 + ω 2 ω ( a 2 + ω 2 -Shift δ(t a e a f(t F ( u a (t e a e at f(t F ( a 5
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