Nonhomogeneous linear systems of DEs Diagonalization, Variation of - - PowerPoint PPT Presentation

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Diagonalization Variation of parameters Nonhomogeneous linear systems of DEs Diagonalization, Variation of Parameters ITI 11/04/2020 ITI Nonhomogeneous systems Diagonalization Variation of parameters Diagonalization 1 Variation of

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Diagonalization Variation of parameters

Nonhomogeneous linear systems of DE’s

Diagonalization, Variation of Parameters ITI 11/04/2020

ITI Nonhomogeneous systems

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SLIDE 2

Diagonalization Variation of parameters

1

Diagonalization

2

Variation of parameters

ITI Nonhomogeneous systems

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SLIDE 3

Diagonalization Variation of parameters

Consider a nonhomogeneous system system of first-order DE’s x′ = P(t)x + g(t) The general solution can be expressed as x(t) = c1x(1)(t) + · · · + cnx(n)(t) + v(t) Here c1x(1)(t) + · · · + cnx(n)(t) is the general solution of the homogeneous system x′ = P(t)x and v(t) is a particular solution of the nonhomogeneous system.

ITI Nonhomogeneous systems

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SLIDE 4

Diagonalization Variation of parameters

Consider a nonhomogeneous system system of first-order DE’s x′ = P(t)x + g(t) The general solution can be expressed as x(t) = c1x(1)(t) + · · · + cnx(n)(t) + v(t) Here c1x(1)(t) + · · · + cnx(n)(t) is the general solution of the homogeneous system x′ = P(t)x and v(t) is a particular solution of the nonhomogeneous system. There are several methods for determining v(t).

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Consider a system with constant coefficients x′ = Ax + g(t) and let A be a diagonalizable matrix. Consider the matrix of eigenvectors T = (ξ(1), . . . , ξ(n))

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Consider a system with constant coefficients x′ = Ax + g(t) and let A be a diagonalizable matrix. Consider the matrix of eigenvectors T = (ξ(1), . . . , ξ(n)) Define new variables y = T −1x, x = Ty

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Consider a system with constant coefficients x′ = Ax + g(t) and let A be a diagonalizable matrix. Consider the matrix of eigenvectors T = (ξ(1), . . . , ξ(n)) Define new variables y = T −1x, x = Ty In the new variables the system becomes Ty′ = ATy + g(t) ⇒ y′ = T −1ATy + T −1g(t) y′ = Dy + h(t), h(t) = T −1g(t)

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Since D is a diagonal matrix, the new system of DE’s is a system

  • f uncoupled equations

y′

i (t) = λiyi(t) + hi(t)

ITI Nonhomogeneous systems

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SLIDE 9

Diagonalization Variation of parameters

Since D is a diagonal matrix, the new system of DE’s is a system

  • f uncoupled equations

y′

i (t) = λiyi(t) + hi(t)

Using the method of integrating factors we find that the solutions are yi(t) = eλit

  • e−λishi(s) ds + cieλit

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Since D is a diagonal matrix, the new system of DE’s is a system

  • f uncoupled equations

y′

i (t) = λiyi(t) + hi(t)

Using the method of integrating factors we find that the solutions are yi(t) = eλit

  • e−λishi(s) ds + cieλit

The solution in terms of the original variables is obtained by changing the variables back x(t) = Ty(t)

ITI Nonhomogeneous systems

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SLIDE 11

Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

ITI Nonhomogeneous systems

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SLIDE 12

Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

The eigenvalue - eigenvector pairs are λ1 = −3, ξ1 =

  • 1

−1

  • λ2 = −1, ξ2 =

1 1

  • ITI

Nonhomogeneous systems

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SLIDE 13

Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

The eigenvalue - eigenvector pairs are λ1 = −3, ξ1 =

  • 1

−1

  • λ2 = −1, ξ2 =

1 1

  • The change of variables matrix is

T =

  • 1

1 −1 1

  • , T −1 = 1

2 1 −1 1 1

  • , y = T −1x

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

The eigenvalue - eigenvector pairs are λ1 = −3, ξ1 =

  • 1

−1

  • λ2 = −1, ξ2 =

1 1

  • The change of variables matrix is

T =

  • 1

1 −1 1

  • , T −1 = 1

2 1 −1 1 1

  • , y = T −1x

In the new variables the system reads y′ = Dy + T −1g(t) = −3 −1

  • y + 1

2 2e−t − 3t 2e−t + 3t

  • ITI

Nonhomogeneous systems

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Diagonalization Variation of parameters

In components the equations are y′

1 + 3y1 = e−t − 3

2t, y′

2 + y2 = e−t + 3

2t and the solutions are y1 = e−t 2 − t 2 + 1 6 + c1e−3t, y2 = te−t + 3t 2 − 3 2 + c2e−t

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

In components the equations are y′

1 + 3y1 = e−t − 3

2t, y′

2 + y2 = e−t + 3

2t and the solutions are y1 = e−t 2 − t 2 + 1 6 + c1e−3t, y2 = te−t + 3t 2 − 3 2 + c2e−t Changing the variables back we have x = Ty =

  • 1

1 −1 1 y1 y2

  • x(t) = c1e−3t
  • 1

−1

  • + c2e−t

1 1

  • + 1

2e−t

  • 1

−1

  • +te−t

1 1

  • + t

1 2

  • − 1

3 4 5

  • ITI

Nonhomogeneous systems

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Diagonalization Variation of parameters

Consider again the nonhomogeneous system x′ = P(t)x + g(t). Assume that a fundamental matrix Ψ(t) for the homogeneous system x′ = P(t)x has been found. Then the general solution for the homogeneous system is x(t) = Ψ(t)c.

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Consider again the nonhomogeneous system x′ = P(t)x + g(t). Assume that a fundamental matrix Ψ(t) for the homogeneous system x′ = P(t)x has been found. Then the general solution for the homogeneous system is x(t) = Ψ(t)c. An ansatz for the nonhomogeneous system x(t) = Ψ(t)u(t)

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Consider again the nonhomogeneous system x′ = P(t)x + g(t). Assume that a fundamental matrix Ψ(t) for the homogeneous system x′ = P(t)x has been found. Then the general solution for the homogeneous system is x(t) = Ψ(t)c. An ansatz for the nonhomogeneous system x(t) = Ψ(t)u(t) Plugging into the equation we have

✘✘✘✘ ✘

Ψ′(t)u(t) + Ψ(t)u′(t) = ✭✭✭✭✭✭

✭

P(t)Ψ(t)u(t) + g(t) u′(t) = Ψ−1(t)g(t) ⇒ u(t) =

  • Ψ−1(t)g(t) dt + c

x(t) = Ψ(t)c + Ψ(t)

  • Ψ−1(t)g(t) dt

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

ITI Nonhomogeneous systems

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SLIDE 21

Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

The eigenvalue - eigenvector pairs are λ1 = −3, ξ1 =

  • 1

−1

  • λ2 = −1, ξ2 =

1 1

  • ITI

Nonhomogeneous systems

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SLIDE 22

Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

The eigenvalue - eigenvector pairs are λ1 = −3, ξ1 =

  • 1

−1

  • λ2 = −1, ξ2 =

1 1

  • A fundamental matrix and its inverse are

Ψ(t) =

  • e−3t

e−t −e−3t e−t

  • , Ψ−1(t) = 1

2 e3t −e3t et et

  • ITI

Nonhomogeneous systems

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SLIDE 23

Diagonalization Variation of parameters

Let’s consider an example x′ = −2 1 1 −2

  • x +

2e−t 3t

  • = Ax + g(t)

The eigenvalue - eigenvector pairs are λ1 = −3, ξ1 =

  • 1

−1

  • λ2 = −1, ξ2 =

1 1

  • A fundamental matrix and its inverse are

Ψ(t) =

  • e−3t

e−t −e−3t e−t

  • , Ψ−1(t) = 1

2 e3t −e3t et et

  • The DE for the coefficient u is

u′(t) = Ψ−1(t)g(t) = e2t − 3/2te3t 1 + 3/2tet

  • ITI

Nonhomogeneous systems

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Diagonalization Variation of parameters

The DE’s for the two components of u and their solutions are u′

1 = e2t − 3

2te3t ⇒ u1(t) = 1 2e2t − 1 2te3t + 1 6e3t + c1 u′

2 = 1 + 3

2tet ⇒ u2(t) = t + 3 2tet − 3 2et + c2

ITI Nonhomogeneous systems

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Diagonalization Variation of parameters

The DE’s for the two components of u and their solutions are u′

1 = e2t − 3

2te3t ⇒ u1(t) = 1 2e2t − 1 2te3t + 1 6e3t + c1 u′

2 = 1 + 3

2tet ⇒ u2(t) = t + 3 2tet − 3 2et + c2 Putting everything together we obtain x(t) = Ψ(t)u(t) x(t) = c1e−3t

  • 1

−1

  • + c2e−t

1 1

  • + 1

2e−t

  • 1

−1

  • +te−t

1 1

  • + t

1 2

  • − 1

3 4 5

  • ITI

Nonhomogeneous systems