Finding the amplitude an oscillator: Driven harmonic oscillator problem

[Benzoate]

Homework Statement

A car is moving along a hill at constant speed on an undulating road with profile h(x) where h'(x) is small. The car is represented by a chassis which keeps contact with the road , connected to an upper mass m by a spring and a damper. At time t, the upper mas has displacement y(t) satisfies a differential equation of the form

y(double dot) + 2Ky(single dot)+\Omega²= 2Kch'(ct) + \Omega²h(ct)

where K and \Omega are positve constants.

Suppose that the profile of the road surface is given by h(x) = h₀cos(px/c), where h₀ and p are positive constants. Find the amplitude a of the driven oscillations of the upper mass.

I will post the website that contains my professor's hint to this problem and since the hint is in pdf form, I am unable to paste it

http://courses.ncsu.edu/py411/lec/001/

Go to homework tab
Then go to assignment 7
then go to 5.11 once you've clicked on assignment 7

Homework Equations

The Attempt at a Solution

y(double dot)+2K\varsigma'+\Omega²\varsigma=0

y=h(ct)+\varsigma==> \varsigma=y-h(ct)

\varsigma(single dot)=y(single dot)-h'c(ct)
y=ce^ipt
y(single dot)=cipe^ipt
y(double dot)=-c^2e^ipt
since h(x)=h(ct) and h(x) = h₀cos(px/c),then h(x)= h₀cos(px/c)= h₀cos(pt)

h(x)= h₀cos(pt)
h'(x)=-p h₀sin(pt)

could I say h(x)= h₀cos(pt)=h0e^ipt?

then
h(x)=h0e^ipt
h'(x)=iph0e^ipt

therefore, \varsigma=y-h(ct) becomes \varsigma=y-h(pt)=> \varsigma(single dot)=y(single dot)-h'p(pt)

plugging all of my variables into the equation y(double dot) + 2Ky(single dot)+\Omega²= 2Kch'(ct) + \Omega²h(ct)

I find c to be :

c=h0(2kp^2+\Omega²)/(\Omega²+2ki-2kh0p)

I do realize in order to get the amplitude I have to calculate the magnitude of c: I think I calculated my magnitude incorrectly :

According to my textbook , here is the actually amplitude

a= ((\Omega⁴+4K^2p^2)/((\Omega²-p^2)²+4K^2p^2))^1/2

 

[Benzoate]

anybody have any trouble reading my solution?