A. Use the method of variation of parameters to determine the general solution of the given differ- ential equations: 1. y" + y' = tan t, -7/2 < t < n/2

4. Find / 4 (36x? +20x) 7 6r+Sx° dx.

What is the slope of the curve y O-1 1 -2 2 = 2 cos x at x = 2

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.

Find the equation of the tangent line to the curve 2 sin a at the point (1). 6 The equation of this tangent line can be written in the form y = mx + b where m = and b