Second order linear differential equations

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SUMMARY

The discussion focuses on solving second-order linear differential equations, specifically the equations ẍ + 5ẋ + 4x = 0 and ẍ + x = cos(t). The first equation demonstrates overdamping, with the solution derived as x(t) = CE-4T + DE-t, where C = 1/5 and D = 1/5. For the second equation, participants suggest using a similar approach to find the homogeneous solution and a particular integral for the steady-state solution.

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CloudKel
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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)
 
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CloudKel said:

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 xh

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

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