2nd derivative equation question?

Question

please help me with this question

determine a particular solution of my'' + cy' + ky = Fcos(wt) of the form Acos(wt - homogeneous solution)

show that the amplitude A is a maximum when w^2 = k/m - 0.5(c/m)^2

thanks so much

Answer 1

particular solution
y=a cowt +bsinwt
y´= -awsinwt +bwcoswt
y´´=-aw^2 cos wt -bw^2 sin wt
Fcoswt = cos wt( -maw^2+cbw+ka) +sinwt( -mbw^2-caw+kb)
so
a(k-mw^2)+b(cw)=F
-acw+b(k-mw^2)=0 These equation allow you to calculate a and b
with these values
define a= A cos @ and b=A sin @
so A^2= sqrt(a^2+b^2) and tan@= b/a
and the particular solution becomes
y=A cos(wx-@)