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USC / Math / MATH 245 / What is the meaning of undetermined coefficients?

What is the meaning of undetermined coefficients?

What is the meaning of undetermined coefficients?

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School: University of Southern California
Department: Math
Course: Mathematics of Physics and Engineering 1
Term: Fall 2016
Tags: Math
Cost: 50
Name: MATH245 study guide midterm 2
Description: midterm 2
Uploaded: 11/03/2016
9 Pages 45 Views 1 Unlocks
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NOTESDon't forget about the age old question of What are the categories of films that share some common qualities?

4.5        x̄ = p̄(t) x̄ + ḡ

        x̄P is its particular solution, {x̄1 , x̄2} is a fundamental set of solutions to its homogenous equation, then a general solution to the non homogeneous solution: x̄ = c1x̄1 (t) + c2x̄2 (t) + x̄p (t)Don't forget about the age old question of Why is it hard to measure the fukushima fallout?

        If Y is any particular solution to y” + p(t) y’ + q(t) y = g(t)

        {y1, y2} is a fundamental set of solutions to its homogenous equation, then a general solution: y = c1 y1 (t) + c2 y2 (t) + y(t)

We also discuss several other topics like What are the point and nonpoint sources of water pollution?

Undetermined Coefficients:Don't forget about the age old question of What is a heavy metal poison called?

        The particular solution of ay” + by’ + cy = gi (t)

        
        gi (t)

  • Pn (t) = A0 tn + A1 tn-1 + …. + An
  • Pn (t) e𝞪 t
  • Pn (t) e𝞪 t |sin 𝛃tcos 𝛃t

Don't forget about the age old question of What is receiving in listening process?
We also discuss several other topics like What is the function of cytoplasmic?

        Yi (t)

  • ts(A0tn + A1tn-1 + … + An)
  • ts(A0tn + A1tn-1 + … + An) e𝞪t
  • ts [(A0tn + A1tn-1 + … + An) e𝞪t cos𝛃 t
  • (Botn + B1tn-1 + ....+ Bn) e𝞪t sin𝛃 t]

4.6        The steady-state response to my” + ry’ +ky = Aeiwt

        Y = A | G(iw) | ei(wt - Φ(wi))

        G (iw) is called frequency response of system

        | G (iw) | and Φ (w) are gain and the phase of frequency response.

        Frequencies at which | G (iw) | are sharply peaked are called resonant frequencies of system

Procedure for Finding the General Solution of x̄ = Ā x̄ when Ā has complex

Eigenvalues:

  1. Identify the complex conjugate eigenvalues λ = μ ± i v
  2. Determine the eigenvector v̄ = (V1V2) corresponding to λ1 = μ ± i v by solving
                    (Ā - λ1Ī ) v̄ = 0
  3. Express the eigenvector v̄ in the form v̄ = ā + b̄ <

  1. Write the solution v̄1 corresponding to v̄ and separate it into real and imaginary parts.

        ȳ1 (t) = eμt (ā cos vt - b̄ sin vt) + i eμt (ā sin vt + b̄ cos vt)

  • eμt (ā cos vt - b̄ sin vt): ū(t)
  • eμt (ā sin vt + b̄ cos vt): w̄(t)

        

It can be shown that ū and w̄ form a fundamental set of solutions for x̄’ = Ā * x̄

  1. Then the general solution of x̄’ = Ā * x̄ is
            x̄(t) = c1 ū(t) + c2 w̄(t)
    where c1 and c2 are arbitrary constants
            eivt = cos (vt) + i sin(vt)

For two-dimensional system with real coefficients, we have now completed our description of the three main cases that can occur:

  1. Eigenvalues are real and have opposite signs: x̄ = 0 saddle
  2. Eigenvalues are real and have the same sign but are unequal: x̄ = 0 node
  3. Eigenvalues are complex with nonzero real part: x̄ = 0 spiral point

y = c1 y1 (t) + c2 y2 (t) is a general solution of y” + py’ + qy = 0 if and only if x̄ = c1 (y2 / y’2) + c2 (y2 / y’2 ) is a general solution of x̄’ = ( -q(t)0  -p(t)1 ) x̄

4.3

        

The form of a general solution ay” + by’ + cy = 0, a ≠ 0, depends on the roots:

                

        of the characteristic equation aλ2 + bλ + c = 0

  • Real and distinct roots if b2 - 4ac > 0,
            λ1 , λ2 are real and distinct, therefore, y = c1 eλ1t + c2 eλ2t
  • Repeated roots if b2 - 4ac = 0,
            λ = λ1 = λ2 = -(b/2a) , therefore,
            y = c1 eλ1t + c2 eλ2t
  • Complex roots if b2 - 4ac < 0,

        λ1 = μ + iv                λ2 = μ + iv

        therefore, y = c1 eμt cos vt + c2 eμt sin vt

x̄’ = Ā x̄ = ([-c/a]0 [-b/a]1] x̄         x̄ = (y’y)

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