18. 03SC Practice Problems 27 Laplace transform Solution suggestions 1. Use e t t2 2 1 is L t2 1 s23 1 2. ss3 So, by s -shift, the transform of the entire expr ession is L e t t2 1 2 s 1 3 s 2s 3. s 1 3.

 

18. 03SC Undetermined cients Solution 1. Find the polynomial solution of x¨ x t2 t 1. 2 1 is nonzero , we can use the method of unde termined coef cients to solve by guessing a quadratic.

 

18. 03SC Practice Problems 8 Exponential and Sinusoidal Input Solution Suggestions 1. Find a solution of x 2x et of the form wet. Do the same for z 2z e2i t First we want to nd a w so that wet satis.

 

18. 03SC Practice Problems 34 T Plane Solution suggestions 1. On the plane, where can you guarantee that any matrix with that value of trace and determinant.

 

18. 03SC Practice Problems 21 Fourier Series: Introduction This pr oblem session is intended as pr eparation for working with Fourier series. 1. What.

 

18. 03SC Matrix The equation u Au or the matrix A is “stable” if all solutions tend to 0 as t ¥. ! “unstable” if most solutions grow without bound as t ¥. !“neutrally stable” otherwise.

 

18. 03SC Practice Problems 26 Convolution Convolution product: The convolution product of two functions f t and g t is Z t f g t f t 0 This We t 0. q t. 1. a Compute t 1. More , q 1 t in q q t. b Compute q t in q q t. Your answers should.

 

18. 03SC Practice Problems 6 Complex numbers 1. n iq Then zn for n 0, 2, 3, 4. 2. a bi 1 p3n a bi n a bi ea bi n. How n 0, 1, 2, 3, 4.

 

18. 03SC Practice Problems 8 Exponential and Sinusoidal Input 1. Find a solutionof x tt2it 2x e of the form we. Do the samefor z 2z e. First determine the appropriate corresponding form.

 

18. 03SC Practice Problems 14 Frequency Response We will be inter ested in a sinusoidal input signal, Fext t A cos wt , and in the steady state, also sinusoidal,.

 

18. 03SC Practice Problems 27 Laplace transform Rules for the Laplace transform Z ¥ L f t F s f t e st dt for Re s 0. 0 Linearity: L af t bg t aF s bG s. L 1L ert f t F s r. s -derivative ule: L tf t F0 s. t-derivative ule: L f 0 t sF s f 0. Formulas for

 

18. 03SC Practice Problems 12 Gain and phase lag Exponential response formula ERF : A solution to p D x Aeis given by ert xp Ap r , p r 6 0. rt is given tert by xp Ap0 r 6 0. 1. Explain. 2. Consider mx¨ bx kx ky. jHj.

 

18. 03SC Practice Problems 9 Solutions to second order ODEs 1. dif fer ential x¨ w2x 0 This harmonic oscillator. 2. f , x¨ w2x 0. 3. f , which have x 0 0 Doesn t this contradict.

 

18. 03SC Practice Problems 1 Introduction, natural growth and decay, review of logarithm 1, and there is assumed a constant harvesting rate of a oryxes/year. Tasks:.

 

18. 03SC Practice Problems 4 Linear models A oblem 1. 2. Now assume that c and r ar e constant. In fact, suppose that r 2 liters/minute and the volume of the tank is V 1 liter.

 

18. 03SC Practice Problems 28 Inverse Laplace transform Rules for the Laplace transform De nition: L f t F s f t e st dt for Re s 0. 0 Linearity: LL 1: F s essentially f t for t 0. s -shift ule: L ert f t F s r. s -derivative ule:.

 

18. 03SC Practice Problems 24 Step and delta functions 0 for t 1 1. Let Q t 2t 2 for 1 t 2 2t 1 for 2 t 3 5 for 3 t a Sketch Is b Find q t Q0 t , c Describe a scenario which might be modeled by the equation x kx q t your k est d Describe.

 

Linear phase portraits The matrices I want you to study all have the form A a 2 2 1. 1. Compute the trace, determinant, c ha racteristic polynomial, and eigen v alues, in terms.