1.6. FORCING TERMS: RESONANCES 30 25 25 20 20 15 시 10 5 0 α=0.02 α=0.01 α=0.00 115 0 0.2 0.4 0.6 0.8 1 w Figure 1.26: The ratio of output to input amplitudes w/VD as a function of the forcing frequency w for several choices of the damping factor a. 6. The Wronskian of two solutions Y₁(a) and Y2(x) to the homogeneous equation (1.6.34) is dy₁ dx W(x) = Y₁(x) x 2 (x) – Y2(x) (x). dY2 da Show that it satisfies the equation, - dW dx -(x) = -a W(x). Solve this equation with W(0) = Wo, and show that W(x) is never zero if Wo 0. What conclusion can you draw from this result?

1) The position of an object at time tt is given by r(t)r(t). Find the velocity, speed, and acceleration both as functions of tt and at the indicated value of tt.r(t)=⟨3sin(t),4cos(t)⟩ , t=π4   2) Find the position function for the object with the given acceleration, initial velocity, and initial position.a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, Position Function

Find L{f(t)} using the First Shifting Theorem. Note: a, b, e, w, k and n are constants. 1. -kt 2. L{e (acost+ 6 sin t)}

3. The table below shows the highest daily temperatures (in degrees Fahrenheit) averaged over the month for the city of Binghamton, NY. Assume Jan represents t = 1. Month Jan Feb Mar Apr May Мay Jun Jul Aug Sep Oct Nov Dec Тemp. | 29 32 41 54 65 74 78 76 68 57 45 33 a) Use the graphing calculator to find an approximate equation of f (x) in the form y = a sin(k(x - b)) + c. Round a, b, k, and c to four decimal places. b) Using the model from a), what months will the temperature exceed 60°F?