Deriving Variation of Parameters for Systems

In summary, the homework statement states that the student needs to derive the method of variation of parameters for scalar equations using a first order system and then convert it into a second order system. The task is to find the fundamental matrix for a system with known fundamental set {y1, y2}.
  • #1
rookie404
3
0
1.Homework Statement
We know the derivation of the method of variation of parameters for second order scalar differential. The task is to derive the method of variation of parameters for scalar equations using this approach: first convert the scalar equation into the first order system and then apply the method of variation of parameters for systems.

Hint: Assume that the fundamental set of solutions {y1(t), y2(t)} for your scalar equation is known. How would you construct a fundamental matrix for your system from the scalar fundamental set?

Homework Equations


(my work, not in problem statement)
y''+p(t)y'+q(t)y=g(t) <--Second order nonhomogenous scalar
y' = P(t)y + g(t) <--first order nonhomogenous system

The Attempt at a Solution


y''+p(t)y'+q(t)y=g(t)
change of variables...

z(t)= [ z1(t)
z2(t) ]

z1(t) = y(t)
z2(t) = y'(t)

z'2(t)+p(t)z2(t)+q(t)z1(t)=g(t)
z'1(t) = y'(t) = z2(t)
z'2(t) = y''(t) = -z2(t)p(t)-z1(t)q(t)+g(t)

P(t) =
[ 0 1
-q(t) -p(t) ]

G(t) =
[ 0
g(t) ]

y' = P(t)y + g(t) <--now what I have has this form (first order system)

Now I have the variation of parameters for systems derivation in front of me, the end result of which is:

y(t) = [tex]\psi[/tex](t)u0 + [tex]\psi[/tex](t)[tex]\int_t_0^t[/tex][tex]\psi[/tex]-1(s)g(s)ds

So I guess the hard part now is finding what the fundamental matrix [tex]\psi[/tex] is. I know the fundamental set is {y1, y2}
In the variation of parameters for second order differential equations we make the assumption y1u'1 + y2u'2 = 0. I am not supposed to make this assumption of this problem.
 
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  • #2
I know the solution involves creating the fundamental matrix, [tex]\phi[/tex]. After that I should just be able to plug it into the variation of parameters equation I already have at the bottom of my 1st post there. I also know that this fundamental matrix is created using my fundamental set which is {y1, y2} for the scalar second order equation. The last thing I know is that I should be using the Wronskian to help me create this matrix, the Wronskian should be the same for the first order system and second order scalar, I believe?
Now where do I go next? Any help is GREATLY appreciated, I have never received help from this forum before, but I have heard good things about it.
Thanks in advance!
 
  • #3
Anyone? Anything? Any help would be appreciated.
 

FAQ: Deriving Variation of Parameters for Systems

1. What is the purpose of deriving variation of parameters for systems?

The purpose of deriving variation of parameters for systems is to find a particular solution to a non-homogeneous linear differential equation. This method is used when the coefficients of the equation are not constant, making it difficult to find an explicit solution.

2. How does the variation of parameters method work?

The variation of parameters method involves finding a particular solution by assuming it has the same form as the homogeneous solution, but with undetermined coefficients. These coefficients are then solved for using a system of equations determined by substituting the particular solution into the original non-homogeneous equation.

3. What are the advantages of using variation of parameters over other methods?

One advantage of using variation of parameters is that it can be used for a wider range of non-homogeneous linear differential equations, including those with variable coefficients. It also does not require the use of integrating factors or undetermined coefficients, making it a more straightforward method.

4. Are there any limitations to the variation of parameters method?

One limitation of the variation of parameters method is that it can only be used for first-order and second-order linear differential equations. It also requires finding the homogeneous solution before being able to find the particular solution, which may be difficult for some equations.

5. Can the variation of parameters method be used for systems of differential equations?

Yes, the variation of parameters method can be extended to systems of differential equations. The only difference is that the particular solution will be a vector instead of a single function, and the coefficients will be solved for using a system of equations with matrices.

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