Steady State Solution for the Equation 4y''+4y'+17y=202cos3t

In summary, the conversation discusses the steady state solution to the equation 4y''+4y'+17y=202cos3t (*), which is found to be Kcos3x+Msin3x. The conversation also explores different ways of expressing the steady state solution, such as Ccos(ωt-η) and K cos(A+B). The formulas K=sqrt(C^2+D^2) and tan B =D/C are suggested as possible methods for finding B and K given C and D.
  • #1
kasse
384
1

Homework Statement



What is the steady state solution to the equation

4y''+4y'+17y=202cos3t (*) ?

2. The attempt at a solution

The steady state solution is, if I've got it right, only the particular solution. It's got to be on the form

Kcos3x+Msin3x.

I calculate the derivatives, put into (*) and find it to be 122,19 cos 3t + 210 sin 3t.

It's normal to write steady state solutions on the form

Ccos(ωt-η).

How can I do that?
 
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  • #2
If I tell you that K cos(A+B) = K cos A cos B - K sin A sin B = C cos A + D sin A,

could you find B and K if you were given C and D ?
 
  • #3
genneth said:
If I tell you that K cos(A+B) = K cos A cos B - K sin A sin B = C cos A + D sin A,

could you find B and K if you were given C and D ?

K=sqrt(C^2+D^2)

tan B =D/C

Found some formulas on the internet. Is it correct?
 
Last edited:

1. What is the Steady State Solution for the given equation?

The Steady State Solution for the equation 4y''+4y'+17y=202cos3t is a particular solution that stays constant over time, despite any changes in the initial conditions or external forces. It is found by setting the derivative terms to zero and solving for the remaining variable.

2. How is the Steady State Solution different from the general solution?

The general solution includes all possible solutions to the given equation, while the Steady State Solution is a specific solution that remains constant over time. The general solution also includes the complementary solution, which accounts for any transient behavior in the system.

3. How can the Steady State Solution be used in practical applications?

The Steady State Solution is useful for analyzing and predicting the behavior of a system over time. It can be used in various fields such as physics, engineering, and economics to understand the long-term behavior of a system under specific conditions.

4. Can the Steady State Solution change over time?

No, the Steady State Solution remains constant over time as it is not affected by any changes in initial conditions or external forces. It represents the equilibrium state of the system.

5. What are some limitations of using the Steady State Solution?

One limitation is that it assumes the system is in a steady state, which may not always be the case in real-world situations. Additionally, it only accounts for linear systems and may not accurately predict the behavior of non-linear systems.

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