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    Probleme Curs 9

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    Sisu Octavian
    This document discusses solving systems of linear differential equations with constant coefficients. It contains problems involving finding the solutions of systems of differential equations given initial conditions. It also discusses using power series representations to solve systems of differential equations. Finally, it covers properties and calculations involving the exponential of a matrix, including using the matrix exponential as a fundamental matrix of solutions to solve initial value problems for systems of differential equations.

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    Probleme Curs 9

    Uploaded by

    Sisu Octavian
    25 views2 pages
    This document discusses solving systems of linear differential equations with constant coefficients. It contains problems involving finding the solutions of systems of differential equations given initial conditions. It also discusses using power series representations to solve systems of differential equations. Finally, it covers properties and calculations involving the exponential of a matrix, including using the matrix exponential as a fundamental matrix of solutions to solve initial value problems for systems of differential equations.

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    This document discusses solving systems of linear differential equations with constant coefficients. It contains problems involving finding the solutions of systems of differential equations given initial conditions. It also discusses using power series representations to solve systems of differential equations. Finally, it covers properties and calculations involving the exponential of a matrix, including using the matrix exponential as a fundamental matrix of solutions to solve initial value problems for systems of differential equations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    25 views2 pages

    Probleme Curs 9

    Uploaded by

    Sisu Octavian
    This document discusses solving systems of linear differential equations with constant coefficients. It contains problems involving finding the solutions of systems of differential equations given initial conditions. It also discusses using power series representations to solve systems of differential equations. Finally, it covers properties and calculations involving the exponential of a matrix, including using the matrix exponential as a fundamental matrix of solutions to solve initial value problems for systems of differential equations.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    VIII.

    Sisteme de ecuatii diferentiale liniare cu coeficienti


    constanti
    1. Sa se determine solutiile urmatoarelor probleme Cauchy:
     0
    x = 5x + 2y
    (a) , x(0) = 3, y(0) = 0.
    y 0 = x 6y
     0
    x = 5x y
    (b) , x(0) = 0, y(0) = 1.
    y 0 = x + 3y
     0
    x =xy
    (c) , x(0) = 1, y(0) = 2.
    y 0 = 4x + y
    0
    x = 3x y + z
    (d) y0 = x + y + z , x(0) = y(0) = z(0) = 2.
    0
    z = 4x y + 4z

    0
    x = 3x y 3z

    (e) y 0 = 6x + 2y + 6z , x(0) = 1, y(0) = 1, z(0) = 0

    0
    z = 6x 2y 6z
     0
    x = 4x + y
    (f) , x(0) = 1, y(0) = 1.
    y 0 = x 2y + e3t
     0
    x = x + y cos t
    (g) , x(0) = 1, y(0) = 2.
    y 0 = 2x y + sin t + cos t
     0
    x = 2x + y + 9t
    (h) , x(0) = 0, y(0) = 1.
    y 0 = x + 4y
     0
    x = 2x y + sin t
    (i) , x(0) = 1, y(0) = 4.
    y 0 = 4x + 2y + cos t
     0
    x = 2x + y + 4et
    (j) , x(0) = 0, y(0) = 1.
    y 0 = 3x + 4y
     0
    x = x + 2y + 3
    (k) , x(0) = 3, y(0) = 2.
    y 0 = 2x + y
     0
    x = 3x y
    (l) ,x(0) = 2, y(0) = 1.
    y 0 = x y + 4e2t
     0
    x = x + y + e2t sin t
    (m) , x(0) = 1, y(0) = 21 .
    y 0 = 2x + 3y e2t cos t
     0
    x = 2x y + 4t
    (n) , x(0) = 0, y(0) = 1.
    y 0 = x + 2y 5t2
     0
    x = 5x 4y + 1
    (o) , x(0) = 1, y(0) = 1.
    y0 = x + y
     0
    x = x 4y + cos 2t
    (p) , x(0) = 1, y(0) = 1.
    y0 = x + y

    2. Rezolvati urm
    atoarele sisteme dePecuatii diferentiale, cautand solutiile sub forma unor serii de
    puteri x(t) = n0 an tn , y(t) = n0 bn tn :
    P
    (
    x0 = 6x 4y
    (a) , x(0) = 1, y(0) = 1
    y 0 = 9x1 6y
    (
    x0 = 6x 4y + et
    (b) , x(0) = 7, y(0) = 9
    y 0 = 9x 6y

    R
    3. Exponentiala unei matrici. Fie A Mn ( ). Matricea
    1 1 X 1
    eA = In + A + A2 + . . . = An
    1! 2! n0
    n!

    se numeste exponentiala matricei A. Se poate arata ca seria de mai sus este convergenta pentru
    orice matrice A Mn ( ). R
    (a) Aratati ca:
    i. e0n = In .
    R
    ii. eA+B = eA eB daca A, B Mn ( ) si AB = BA.
    R
    iii. e este o matrice inversabila pentru orice A Mn ( ), si eA = eA .
    A 1

    iv. Functia R R
    Mn ( ), t 7 eAt este derivabila, si (eAt )0 = AeAt = eAt A (n particular,
    rezulta ca matricile A si eAt comuta).
    (b) Se poate arata ca solutia generala a sistemului liniar omogen de ecuatii diferentiale

    X 0 = AX (SO)

    este
    X(t) = eAt C unde C Rn
    Rezulta ca M(t) = eAt este o matrice fundamentala de solutii. In particular, solutia
    problemei Cauchy (
    X 0 = AX
    X(t0 ) = X0
    este
    X(t) = eA(tt0 ) X0
    R
    (c) Calculul matricei exponentiale. Fie A Mn ( ) o matrice care admite o descompunere de
    R
    tipul A = P D P 1 , unde P, D Mn ( ), P inversabila. Aratati ca

    eAt = P eDt P 1
       t 
    1 0 Dt e 1 0
    i. Daca D = , verificati ca e = .
    0 2 0 e 2 t
       t 
    1 Dt e tet
    ii. Daca D = , verificati ca e = .
    0 0 et
       at at
    
    a b e cos bt e sin bt
    iii. Daca D = , verificati ca eDt = .
    b a eat sin bt eat cos bt
    (d) Reluati exercitiul 1 folosind matricea exponentiala ca matrice fundamentala de solutii.

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