Test2Solutions - Math 306 — Test 2 Printed Name Please...

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Unformatted text preview: Math 306 — Test 2 Printed Name: Please carefully work all of the following problem(s). You must SHOW YOUR WORK to receive ANY credit! The Printed Name question is worth 5 points. a 'H indicates a “By Hand” problem where technology is not allowed. 0 '1? indicates a “Technology” problem where you w use Mathematica or a calculator. You must still .5 show the steps taken. For example, / (1 — I)? dm 2 w8,192 is valid. Giving the answer —8, 192 Without 1 the integral is not. I Do not use DSolve or NDSolve unless otherwise directed (except for checking your work). Show that y1(a:) m e217 and y2{.r) : e3”: are linearly independent. Solve initial value problem, y”(.r) + 5y’(a:) + Gym) 2 0 where 5(0) 2 —-1, y’(0) = 3. («1 ‘i'lr- M ’39 (r “NH-1‘} :u \ Dr. Emmert Spring 2010 I of III Points Earned [:i of 0 Math 306 w» Test 2 Find the general solution for the differential equation m”(t) + 21%) + = 0. Consider the plot of the solution to a free Inass—spring—dashpot system shown below. What type of vibrations does this system exhibit (Over—damped, Critically Damped, or Under-damped)? Determine the appropriate form (no need to find the constants - just give the “guess” that the Method of Undetermined Coefficients would use!) for a particular solution of y” + 611’ + 13y : (Pam cos(23:) m3m( Note that the complementary solution is 16m) = e c; (103(21') + Cg sin{a:)). u... 61W? U a“ 3N: (till/53E) {WKIIXJJ (911$th Lm‘ibtni) w? Points Earned 1:] of 0 II of III Dz. Emmert Spring 2010 Math 306 — Test 2 E A body with mass 0.25 kg is attached to the end of a. spring that is stretched 0.25 meters by a force of 9 Newtons. At time t z 0 the body is pulled 1 meter to the right, stretching the spring, and set in motion with an initial velocity of 5 meters per second to the left. Construct an Initial Value Problem for this situation. F: bx Walker? €%?+ER 2C7 ng}m-_}f‘ l1: Eli/{fig 96% Max Io .3; 'll' C Plot .17 15 over the interval 0 g t g 3. Attach ’our icture and code. 4 3 13 Find a particular solution of y” + y : tan(:1;). Dr. Ennnert Spring 2010 III of III Points Earned E of 0 Problem 1 mm: Wronskian[ {Exp [2 x] , Exp[3 x] } , :c] max]: e5 x Problem 2 (Check Only) mm: DSolve[{Y"[x] +5y'[x] +5ylx] == 0, H0] == —l. y' [0] == 3}: Y[x]- 3:] 0mm: {{yhc] —> —e"“‘}} Problem 3 (Check Only) mpg}: DSolVE[X"[t3+2x'[1‘-] +X[t] == 0: Mt]: t] 0mm:- {{x[t} «a E“ C[1j+ e“: t: CEE] Problem 5 (Check Only) miss}; DSolve [y' ' [x] + 6 y' [x] + 13 y[x] == Exp[—3 3:] C05 [2 x] . y[x] , x] 0mm: {{yix] «a e"“C[2] {305(2):} +e‘”C[l] Sin[2x] + 1 —— e‘” (Cos[2x] COSHX] +4xSin[2x] +Sin[2x} 511191an 16 2 I TestzMarhematlcaSqutIons.nb Problem 6 I PartA The initial value problem is given by x" + 144 x. = D, .\‘(0) = 1, .\" (0) m —5. I PartB mpg: DSolve[{x"[t] +144x[t] == 0, x[0] == 1, x'[u] :2 m5}, x[t], 1;] mum: {{x[t} —> El»; (12 (20.3[12 t] — 5 sin[12 t])}} mus]: c: Sqrt[1‘2 + (—5/12)‘2] l3 SMILE}: W 12 angina: Ref = N[ArcTan[5 / 12]] mama: G . 394791 Infill}? 2 Pi '- Raf mam]: 5.88839 I PartC N11121:: Plot[13 / 12 * Cos[12 t — 5. 88839] , (t, 0. Pi / 4} , Plotstyle —) {Thick, Blue}] Oul[12f= TestZMa mama ticaSolutlons. nb I 3 Problem 7 In Check the solution to the homogenous equation: iral‘Elijtz Dsolve {Y' I [x] ‘1‘ YDCE 3 or Y[x] r 3‘] During: {{y[x] —> C[1] Cos[x} +C[2} Sin[x]}} I: Now, use Variation of Parameters: y1 = Cos [x] ; y: = Sin[x]: f = Tan[x]; W = WronskianHyl, y2} , 3:] Emma IE: 1 mime yp = Simp1i£y[—yl IntegrateEyz f l W, x] + 3:2 Integrate[y1 f / w. x]] auiggrjiz CosEx] [Log[C05[;:] —Sin[:H —L0g[Cos[:m] +Sin[:»H} Notice that you can write the above as [mucosa -sm[:~n/ [use] margin] i ("J 0 [fl L—l ¢_...a CluiiBE]: Cos [x] Log[ w|3 u|$< —Sin[ +Sin[ to”: no}: t...‘ Cos[ ...
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