Second order linear differential equations

[CloudKel]

Homework Statement

a) ẍ + 5ẋ + 4x = 0
x(0) = 0, ẋ(0)=1
What type of damping?

b) ẍ + x = cos(t)
x(0) = 0, ẋ(0)= 1
What type of motion?

The Attempt at a Solution

a) Let x = R Eᴿᵀ

R = -4
R = -1

x(t) = CE**-4T + DE**-t

C + D = 0
-4C - D = 1

C = 1/5
D = 1/5

And it is over damping

I think this is the right answer for a) but i have no idea how to do b)

 

[rock.freak667]

Homework Statement

a) ẍ + 5ẋ + 4x = 0
x(0) = 0, ẋ(0)=1
What type of damping?

b) ẍ + x = cos(t)
x(0) = 0, ẋ(0)= 1
What type of motion?

The Attempt at a Solution

a) Let x = R Eᴿᵀ

R = -4
R = -1

x(t) = CE**-4T + DE**-t

C + D = 0
-4C - D = 1

C = 1/5
D = 1/5

And it is over damping

I think this is the right answer for a) but i have no idea how to do b)

a is correct, didn't check your values though.

You need to solve b) in the same way.

Set the LHS to zero and solve for x_h

and your steady-state solution would be the PI for the right side.