Finding the Value of c for 4u"+cu'+6u=0

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In summary, in order for the solutions of the equation 4u"+cu'+6u=0 to tend to zero as fast as possible, c must be equal to 1 or the system must be critically damped with \zeta=1. This means that the transient response is as short as possible and that all poles are on the left half of the s-plane. This is important for the solution to converge and for the Fourier transform to exist.
  • #1
mathgirl2007
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Find the value of the constant c so that solutions of the equation 4u"+cu'+6u=0 tend to zero as fast as possible.


I think that c must be positive in order for this to tend towards zero but I cannot figure out what c has to be.


Thanks for your help!
 
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  • #2
What is the characteristic polynomial of this ODE? What are the solutions (in terms of c) of that polynomial? What does that make the general solution for u?
 
  • #3
this is all i was given.
would it be that m=4 lambda=c and k=6?
 
  • #4
Have you studied ODE's yet?? Are you familiar with the term characteristic polynomial?

For example, the characteristic polynomial of [itex]2u''+5u'-3=0[/itex] is [itex] 2 \lambda^2+5\lambda-3=0[/itex] which has roots [itex] \lambda_1= -3[/itex] and [itex]\lambda_2=\frac{1}{2}[/itex] and a general solution of [itex]u(x)=c_1e^{\lambda_1 x} +c_2e^{\lambda_2 x}[/itex]

So, what is the characteristic polynomial of 4u" + cu'+6u=0?
 
  • #5
So the equation would be [itex] 4\lambda^2+C\lambda+6=0[/itex]
 
  • #6
and the roots would be (-C+/- the square root(c^2 - 96))/8
 
  • #7
Yes, so what is the general solution u(x) then?
 
  • #8
u(x) = c1e^(-C + the square root(c^2 - 96))/8)x + C2e^(-C - the square root(c^2 - 96))/8
 
  • #9
Good, now what does it mean for a function to tend to zero? (in terms of the derivative of the function)
 
  • #10
its going to have to be getting infinitely smaller
 
  • #11
Yes, it gets smaller as x-gets larger and so du/dx is negative correct?

What is du/dx of your function?
 
  • #12
im not sure how to find that here. is that the derivative in regards to x?
 
  • #13
im really stuck. would it be that C must b be 1 to tend towards zero?
 
  • #14
Yes, du/dx is the derivative with regard to x...You know what u(x) is, so calculate u'.
 
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  • #15
okay i did that. now where in there can i find what c is?
 
  • #16
Well u' should be as large of a negative number as possible if u tends to zero as fast as possible correct?

In other words you want to find the minumum value of u'(x) with respect to c. The derivative of u'(x) with respect to c should be zero at its minimum, so solve the equation

[tex]\frac{d u'}{dc}=0[/tex]

for c.
 
  • #17
"(the solution) tends to zero as fast as possible"

My interpretation is that the transient response is "as short as possible".
given a second order system [tex]ay''+by'+cy=dx'+ex[/tex] under zero input excitation or [tex]ay''+by'+cy=0[/tex], the response is due to non-zero initial condition(s). In order for the solution to converge (stable system and non oscillating), it requires either
1) a>0,b>0, and c>0, OR
2) a<0, b<0, and c<0
such that all poles are on the left half of s-plane (excluding [tex]j \omega[/tex]) axis. In the other word, the Fourier transform must exist.

since the OP mentioned m, c, and k, I'm assuming she is dealing with a spring-mass-damper system ([tex]my''+cy'+ky=0[/tex] with non-equilibrium initial displacement
), I bet the answer is when [tex]\zeta=1[/tex] or critically damped. Note that when system is critically damped (double real roots), the solution doesn't take the form as other poster mentioned.
 
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  • #18
klondike said:
"(the solution) tends to zero as fast as possible"

My interpretation is that the transient response is "as short as possible".
given a second order system [tex]ay''+by'+cy=dx'+ex[/tex] under zero input excitation or [tex]ay''+by'+cy=0[/tex], the response is due to non-zero initial condition(s). In order for the solution to converge (stable system and non oscillating), it requires either
1) a>0,b>0, and c>0, OR
2) a<0, b<0, and c<0
such that all poles are on the left half of s-plane (excluding [tex]j \omega[/tex]) axis. In the other word, the Fourier transform must exist.

since the OP mentioned m, c, and k, I'm assuming she is dealing with a spring-mass-damper system ([tex]my''+cy'+ky=0[/tex] with non-equilibrium initial displacement
), I bet the answer is when [tex]\zeta=1[/tex] or critically damped. Note that when system is critically damped (double real roots), the solution doesn't take the form as other poster mentioned.

Yes, that makes more sense :0)
 

1. What is the meaning of 'c' in the equation 4u+cu'+6u=0?

'c' represents the coefficient of the first derivative term, cu', in the given differential equation. It is a constant number that is multiplied to the first derivative of the function u(x).

2. How do you solve for 'c' in the equation 4u+cu'+6u=0?

To solve for 'c', we can use the method of undetermined coefficients. This involves plugging in a guessed solution into the equation and solving for 'c'. Another method is to use initial value conditions, where the value of 'c' can be determined by using the initial value of u(x) and its derivative.

3. What is the significance of finding the value of 'c' in this equation?

Finding the value of 'c' allows us to determine the particular solution to the given differential equation. This is important in understanding the behavior and characteristics of the function u(x) that satisfies the equation.

4. Is there a specific method to find the value of 'c' or can it vary?

The method used to find the value of 'c' can vary depending on the type and complexity of the differential equation. Some equations may require the use of advanced mathematical techniques, while others can be solved using basic algebraic manipulations.

5. Can there be multiple values of 'c' that satisfy the equation 4u+cu'+6u=0?

Yes, there can be multiple values of 'c' that satisfy the equation. This is because the coefficient 'c' is just one factor in the overall solution to the differential equation. Other factors such as initial values and boundary conditions can also affect the value of 'c'.

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