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11 August, 18:48

Consider the equation y′′ + 2ay′ + a2 y = 0. Show that the roots of the characteristic equation are r1 = r2 = - a so that one solution of the equation is e-at. b. Use Abel's formula [equation (23) of Section 3.2] to show that the Wronskian of any two solutions of the given equation is W (t) = y1 (t) y′2 (t) - y′1 (t) y2 (t) = c1e-2at, where c1 is a constant. c. Let y1 (t) = e-at and use the result of part b to obtain a differential equation satisfied by a second solution y2 (t). By solving this equation, show that y2 (t) = te-at.

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  1. 11 August, 22:15
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    Step-by-step explanation:

    Given the differential equation

    y'' + 2ay' + a²y = 0 ... (1)

    The auxiliary equation to (1) is

    m² + 2am + a² = 0

    Solving the auxiliary equation, we have

    (m + a) ² = 0

    m + a = 0 twice

    m = - a twice.

    The complimentary solution for the repeated root is

    y = C1e^ (-at) + C2te^ (-at)

    So, obviously, one of the solutions is

    y1 = C1e^ (-at).

    For C1 = 1, we have

    y1 = e^ (-at)

    Let W be the Wronskian of y1 and y2.

    The Wronskian satisfies the first order differential equation

    W' + 2aW = 0

    Solving this, we have

    W'/W = - 2a

    Integrating this

    lnW = - 2at + C3

    W = Ce^ (-2at)

    (Where C = e^C3)

    The Wronskian of y1 and y2 is the determinant

    |y1 ... y2|

    |y1' ... y2'|

    = y1y2' - y1'y2

    Since y1 = C1e^ (-at)

    y1' = - aC1e^ (-at)

    W = C1e^ (-at) y2' - (-aC1e^ (-at)) y2

    But

    W = Ce^ (-2at)

    So, we have

    C1e^ (-at) y2' + aC1e^ (-at) y2 = Ce^ (-2at)

    Divide through by C1e^ (-at)

    y2' + ay2 = C3e^ (-at) ... (2)

    (where C3 = C/C1)

    Now, solving (2), we have multiply by the integrating factor e^ (at) to have

    d (y2e^ (at)) = C3

    Integrating this, we have

    y2e^ (at) = C3t + C4

    y2 = C3te^ (-at) + C4e^ (-at)

    Here, for C3 = 1, and C4 = 0, we have

    y2 = te^ (-at).
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