-sin(t)y''+cos(t)y'+(sin(t)-2csc(t))y=t^2sin^2t?

show that y1t=sint is a solution to complementary equation.

use the method of reduction of order to construct a second solution y2(t) to the complementary equation and show y1t) and y2(t) form a fundamental set of solutions to the complementary eqation.

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  • kb
    Lv 7
    7 years ago
    Favorite Answer

    Note: (-sin t) y'' + 2(cos t) y' + (sin t - 2 csc t) y = t^2 sin t

    should (more than likely) be the DE; otherwise y = sin t is not a homogeneous solution.

    ----------

    Letting y = sin t:

    (-sin t) y'' + 2(cos t) y' + (sin t - 2 csc t) y

    = (-sin t) (-sin t) + 2(cos t) (cos t) + (sin t - 2 csc t) (sin t)

    = 2( sin^2(t) + cos^2(t)) - 2

    = 2 - 2

    = 0.

    So, y = sin t is a solution to the complementary (homogeneous) equation.

    ----------------

    To use Reduction of Order, assume that y = (sin t) z.

    Differentiating yields

    y' = (cos t)z + (sin t)z'

    y'' = (-sin t)z + 2(cos t)z' + (sin t)z''.

    Substituting this into (-sin t) y'' + 2(cos t) y' + (sin t - 2 csc t) y = t^2 sin^2(t) yields

    (-sin t) [(-sin t)z + 2(cos t)z' + (sin t)z''] + 2(cos t) [(cos t)z + (sin t)z'] + (sin t - 2 csc t) ((sin t) z) = t^2 sin^2(t).

    Simplifying:

    -sin^2(t) * z'' = t^2 sin^2(t).

    ==> z'' = -t^2.

    Integrate both sides twice:

    z' = (-1/3)t^3 + A

    z = (-1/12)t^4 + At + B.

    So, a general solution is

    y = [(-1/12)t^4 + At + B] sin t

    ...= (At sin t + B sin t) - (1/12)t^4 sin t.

    ----------------

    Take y2 = t sin t

    (since y = (-1/12)t^4 sin t is a particular solution, having no arbitrary constants).

    We verify that y1 and y2 form a fundamental set of solutions to the complementary

    equation by checking that their Wronskian is nonzero (as a function in t):

    |..sin t......t sin t..|

    |(sin t)'...(t sin t)'| =

    |sin t.........t sin t.......|

    |cos t...sin t + t cos t| = sin^2(t), which is a nonzero function (in t).

    --------------

    I hope this helps!

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  • Anonymous
    7 years ago

    The answer to said question is:

    0.00003046483

    We hope this answer has helped you emotionaly and physically.

    Source(s): my head
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