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midterm2 - Math 415 Midterm 2 Review Ch3.4-3.9 1...

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Math 415 Midterm 2 Review Ch3.4-3.9 1. Homogeneous equation : ay 00 + by 0 + cy = 0 The form of the general solution depends on the solution of the characteristic eqn ar 2 + br + c = 0. (1) two real roots r 1 , r 2 : y = c 1 e r 1 t + c 2 e r 2 t (2) one real root r : y = c 1 e rt + c 2 te rt (3) complex roots λ ± : y = c 1 e λt cos μt + c 2 e λt sin μt ex) 3.4.11, 3.5.11 2.Reduction of order. Find y 2 when y 00 + py 0 + qy = 0 and a solution y 1 are given. Set y 2 = vy 1 . Look at the eqns (27)- (30) on p.171. Note that (30) is actually 1st order equation and usually this will be separable eqn. ex)3.5.25 3. Non-homogeneous equation : ay 00 + by 0 + cy = g ( t ) (1) general solution y = y h + Y (2) Method of undetermined coefficients : From g ( t ), guess the possible form of Y . (i)Consider the sum of terms of all possible derivative of g with undetermined coefficients. (ii)If this contains some homogeneous solution, multiply t or t 2 . (iii)Then plug into the equation, find the value of coefficients. ex) 3.6.13 , 3.6.19 4.Mechanical vibrations : mu 00 + γu 0 + ku = F ( t ) (1) Hooke’s law mg = kL (2) A cos w 0 t + B sin w 0 t = R cos( w o t - δ ) (p.196) (3) Check following concept (i)frequency, period, amplitude for harmonic oscillation (free undamped) (ii)quasi-frequency, quasi-period for free oscillation with underdamping
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