p1061.pdf - ~ The Classical Wave Equation a2—5a...

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Unformatted text preview: / ~\ The Classical Wave Equation a2—5a+6=0 (EX-’2)(C¥-*3):O 01:2, 3 The general solution is y(x) = clez‘ + ale“. Using the boundary conditions gives y : 6162‘ + 6263’ d: : 26‘2” + 3626’“ __ O 0 .._ » ~l..cle +cze 0_2c|+3c2 , 3C2 “IZCI‘l-Cz (‘I=~—§- Combining these results gives a] = -92 2 or '3 = Cl 2 = c2 The solution is y(x) = «3632" + 2e3"i dv C _'_ —— 2y :- dx ' (1 ~ 2 : 0 oz : The solution is y(x) = 2e“. ________________________._————————————-——-— 2—3. Prove that x(t) = cos cot oscillates with a frequency v : w/2JT. Prove that x(t) = A cos wt + B sin wt oscillates with the same frequency, (0/221. The functions cost and sin I oscillate with a frequency of v 2 1/271, since they go through one complete cycle every 2n radians. The functions cos(wt) and sin(a)t) go through a) complete cycles every 271 radians, so they oscillate with a frequency of v 2 (0/221 A linear combination of these functions (for example, A cos wt + B sin cut) will oscillate with the same frequency, cu/27r. __________________________________.._——————-——-—-———-- 24. Solve the following differential equations: dzx , (1x a. 7 + w“x(t) : 0 x(0) : 0: ~(atr : (l) : v1) (11‘ dr dzx 2 V . dx _ b. EMU x(t)=0 x(0):A: Etatr :u) 2 no Prove in both cases that xtt) oscillates with frequency (Ll/271'. 29 ...
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