How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation?

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Jody Walters
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1 How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation? (A) 0 (B) 1 (C) 2 (D) more than 2 (E) it depends or don t know

2 How many of the following are general solutions of x (t) = ω 2 x(t)? B 1 Acos( ωt δ ) C 1 cos( ωt) + B sin( ωt) A Re[exp(i( ω t δ ))] 2 exp( iωt) + C 2 exp( iωt) where B 1, B 2, C 1, C 2, A and δ are constants (A) 1 (B) 2 (C) 3 (D) All of them

3 (A) 0 (B) 1 (C) i (D) i 2 = -1 (E) cannot be simplified / don t know

6 Consider the amplitude A 2 = 2 f ( ω 0 ω ) + 4β ω In the limit as ω goes to infinity, A (A) goes to zero (B) approaches a nonzero constant (C) goes to infinity (D) I don t know!

7 Consider the amplitude A 2 = 2 f ( ω 0 ω ) + 4β ω In the limit as ω goes to 0, A (A) goes to zero (B) approaches a nonzero constant (C) goes to infinity (D) I don t know!

8 A = What is the shape of? 1 ( ω 2 0 ω 2 ) 2 + 4β 2 ω A) B) ω ω C) 1.0 D) ω ω E) None of these looks at all correct?!

9 If you have a damped, driven oscillator, and you increase damping, β, (leaving everything else fixed) what happens to the curve shown? A 2 = 2 f ( ω 0 ω ) + 4β ω Fixed ω! ω 0 (A) It shifts to the LEFT, and the max value increases. (B) It shifts to the LEFT, and the max value decreases. (C) It shifts to the RIGHT, and the max value increases. (D) It shifts to the RIGHT, and the max value decreases. (E) Other/not sure/??? 14

10 If you have a damped, driven oscillator, and you increase damping, β, (leaving everything else fixed) what happens to the curve shown? A 2 = 2 f ( ω 0 ω ) + 4β ω Fixed ω 0 ω (A) It shifts to the LEFT, and the max value increases. (B) It shifts to the LEFT, and the max value decreases. (C) It shifts to the RIGHT, and the max value increases. (D) It shifts to the RIGHT, and the max value decreases. (E) Other/not sure/??? 15

11 How many of the following are even functions? I: x II: sin(x) III: sin 2 (x) IV: cos 2 (x) (A) None (B) Exactly one of them (C) Two of them (D) Three of them (E) All four of them!

12 How many of the following are even functions? I: 3x 2-2x 4 II: -cos(x) III: tan(x) IV: e 2x (A) None (B) Exactly one of them (C) Two of them (D) Three of them (E) All four of them!

13 How would you classify this function? (A) Odd (B) Even (C) Neither

14 f (t) = n =0 a n cosnωt + b n sinnωt What can you predict about the a s and b s for this f(t)? f(t) (A) All terms are non-zero (B) The a s are all zero (C) The b s are all zero (D) a s are all 0, except a 0 (E) More than one of the above (or none, or???)

15 f (t) = n =0 a n cosnωt + b n sinnωt What can you say about the a s and b s for this f(t)? f(t) t (A) All terms are non-zero (B) The a s are all zero (C) The b s are all zero (D) a s are all 0, except a 0 (E) More than one of the above, or, not enough info...

16 period T = 2π/ω 0 limit as T gets long f (t) = i nω 0 t c n e n =- f (t) - g(ω )e iω t d(what?) (A) dx (B) dt (C) dω (D) Nothing is needed, just (E) Something else/not sure 0 f (t) g(ω )e iω t

17 Fourier Series f (t) = i nω 0 t c n e n =- Fourier Transforms f (t) = - g(ω)e iω t dω c 1 n = T T i nω 0 t f (t)e dt g(ω) f (t)e i what? dwhat? 0 - (A) - f ( t) e inωt dt (B) - f ( t) e iωt dt (C) inωt f ( t) e dω - (E) Something else/not sure? (D) - f ( t) e iωt dω

19 The Fourier transform of a Gaussian function is again a Gaussian function. What can you say about the width of g(ω) if the width of f(t) increases? (A) The width of g(ω) increases as well (A) The width of g(ω) increases as well (B) The width of g(ω) decreases (C) The width of g(ω) stays the same (D) Don t know

20 Delta functions δ (t)dt What are the units of δ(t)? A) unitless B) time C) (time) -1 D) something else E) What is δ(t-t 0 )?

21 Delta functions What is the Fourier transform of δ(t)? A) δ(t) B) δ(ω) C) a constant D) 0 E) something else

22 Delta functions What is f ( t) δ ( t) dt? A) f(t) B) C) 0 D) f(0) E) something else

23 A) 0 Delta functions 0 2 What is x δ ( x 2) dx? B) 1 C) 2 D) 4 E) something else

24 A) 0 Delta functions 0 2 What is x δ ( 2 x) dx? B) -4 C) -2 D) 2 E) 4

25 A) -4 Delta functions 0 2 What is x δ ( x + 2) dx? B) -2 C) 2 D) 4 E) something else

26 What is the expression for the volume density ρ(r) of a point particle with mass m at the origin? (3) A) B) δ (r) r 1 δ (3) ( r) (3) C) D) E) mδ ( r) δ m (3) ( r) m

27 Separation of variables Say you have two functions f(x) and g(y). f(x) depends on x but not on y. g(y) depends on y but not on x. If f(x) + g(y) = 0 for all x, y, then: (A) Both functions are constants (i.e. they do not depend on x, y at all.) (B) Both functions have to be zero everywhere. (C) Impossible, this equation can never be solved. (D) Not much to conclude, just f(x) = -g(y). (E) Something else / Don t know

28 Semi-infinite plate, with temperature fixed at edges: T=0 T 0 2 T(x, y) = 0 T=0 y=0 x=0 T=f(x) x=l When using separation of variables, so T(x,y)=X(x)Y(y), which variable (x or y) has the sinusoidal solution? (A) X(x) (B) Y(y) (C) Either, it doesn t matter (D) NEITHER, the method won t work here (E) Don t know

Fourier series. Complex Fourier series. Positive and negative frequencies. Fourier series demonstration. Notes. Notes. Notes.

Fourier series. Complex Fourier series. Positive and negative frequencies. Fourier series demonstration. Notes. Notes. Notes. Fourier series Fourier series of a periodic function f (t) with period T and corresponding angular frequency ω /T : f (t) a 0 + (a n cos(nωt) + b n sin(nωt)), n1 Fourier series is a linear sum of cosine