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?

Q.4) Find the position of the hammered string of a piano at any time t, the length of the string is 2 meter and it is fixed at both ends. The wave constant c² = 2, the initial displacement of the string is zero and its initial velocity is given by function: u₂(x,0) = sin (xx)

18. In a region of the country, the population of a particular type of beetle is modeled by the function f(t) = 1300 – 140 sin(2t), where t is measured in years. According to the model what is the maximum beetle population? (а) 1160 (b) 1440 (c) 1300 (d) 1580 (e) None of these

Solve the d.e. dv/dt=9.8-0.196v and use desmos to sketch the family of curves with values of the arbitrary constants [-5,5], use integers only. From the family of curves, sketch the isoclines. Could you please show the sketch for isoclines? Thank you!

6. A toy on an elastic string is attached to the top of a doorway. It is pulled down and released, allowing it to bounce up and down. The length of the elastic string, L centimetres, is modelled by the function L = 110 + 25 cos cos (2t) where t is time in seconds after release. (a) Find the length of the elastic string after 2 seconds. (b)Find the minimum length of the string. (c) Find the first time after release that the string is 85 cm. (d)What is the period of the motion?

A spring has a mass of 10 kg, is underdamped with coefficient β = 5 and its frequency is 0.8 Hz. Spring starts 30 cm from equilibrium position and is released with a velocity of −10 cm / s. Set up, in terms of the spring constant k, an equation and conditions for the position of the mass, in meters.

Find the critical point(s) of the function y = 0 - 2 cos (0) at 0 € [-31,-1]. (Give your answer in the form of a comma-separated list of values. Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if the function has no critical points.) critical point(s): Determine the x-coordinates of the critical point(s) that correspond(s) to a local minimum or a local maximum. (Give your answer in the form of a comma-separated list. Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if the function has no local minimum or local maximum.) local minima: local maxima: Find the intervals on which the function is increasing or decreasing. (Give your answer as an interval in the form (*, *). Use the symbol co for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Øif the interval is empty. Express numbers in exact form. Use…

4. A car supported by a MacPherson strut (shock absorber system) travels on a bumpy road at a constant velocity v. The equation modeling the motion of the car is Tut 80x + 1000x = 2500 cos where r = x (t) represents the vertical position of the cars axle relative to its equilib- rium position, and the basic units of measurement are feet and feet per second (this is actually just an example of a forced, un-damped harmonic oscillator, if that is any help). The constant numbers above are related to the characteristics of the car and the strut. Note that the coefficient of time t (inside the cosine) in the forcing term on the right hand side is a frequency, which in this case is directly proportional to the velocity v of the car. (a) Find the general solution to this nonhomogeneous ODE. Note that your answer will have a term in it which is a function of v.

a. -25 -5 b. -40 3. с. 40 0 d. 25 13. A bug is moving with velocity v(t)=t - 6t +8 for t2 0. Find the total distance traveled by the bug over interval [0,6]. a. 0 b. 8 12 44 Copyright Cengage Learning. Powered by Cognero. Pag d.