Differential Equations Application Problems

Hi

Need help on a few questions:

1) The equation of motion of a mass (on the end of a vertical spring) subject to an "input force" (which causes a forced motion) is

$\displaystyle \frac{d^2y}{dt^2} + 9y = 5cos(3t)$

Find y given that when t=0, y=0 and y' = 0

I get

$\displaystyle y = y_p +y_h$

$\displaystyle y = \frac{5}{6}tsin(3t) - \frac{5}{6}sin(3t)$

book's answer is $\displaystyle y= \frac{5}{6}tsin(3t)$

2) a) A particle moving in a straight line has displacement x, velocity v and acceleration a at time t, where $\displaystyle a=\frac{dv}{dt}=v\frac{dv}{dx}$. These formulas can be used to find v given a. The former is used when v is required in terms of t, the latter for v in terms of x.

b) A particle moves with retardation proportional to its velocity, ie, a = -kv, and has initial velocity $\displaystyle v_0$ and initial displacement 0. Find the velocity (i) in terms of time; and (ii) in terms of the displacement.

How would i do this question?

P.S