Consider the vibrating system described by the initial value problem y\'\'+y=3co
ID: 2830919 • Letter: C
Question
Consider the vibrating system described by the initial value problem
y''+y=3cos(wt), y (0)=1, y ' (0)=1
a. Find the solution for w = 1 and w does not =1
b. Plot the solution y(t) versus t for w= 0.9, ans w=1
I have the answer and it can be easily found with google so please do not just copy and paste some answer I need explanations on how to LEARN the material not just slap a answer down so please approach it as if you were teaching someone how to do this thanks! Answer should not be found with laplace transform. Use variation of parameters I think . The answer should have C1...C2...etc.
Explanation / Answer
y" +y = 3coswt
homogenous equation for the above equation is
y"+y = 0
=>
charecteristic polynomial is
p^2 +p = 0
=>
p(p+1) = 0
=>
p = 0,-1
=>
general solution of homogeneous equation is
y = c1+c2e^(-t)
for particular solution:
let y = acoswt +b sinwt be a solution
=>
y"+y' = (-aw^2+bw)coswt +(-bw^2 -aw)sinwt = 3coswt
=>
-aw^2 +bw = 3, -bw^2 -aw = 0
(a)
w = 1:
=>
a = -1.5, b = 1.5=>
general solution of the equation is
y = c1+c2e^(-t) -1.5coswt + 1.5sinwt
y(0) = 1, y'(0) = 1
=>
c1+c2-1.5 = 1=> c1+c2 = 2.5
1.5-c2= 1 => c2 = 0.5 =>c1 = 2
=>
y = 2+0.5e^(-t) -1.5cost + 1.5sint