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Math Help - Using complex numbers for 2nd order DE

  1. #1
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    Using complex numbers for 2nd order DE

    I can change this equation, using Eulers method, and get the answers using sin and cos, but using complex numbers is apparently faster and we have to do it this way for my professor...so if you could help me figure this out I would appreciate it.

    y''-8y'+25y=3 e^{4x} cos(3x)

    HERE IS WHAT I DID:
    First, I'll solve the homogeneous case to find the complementary solution and the roots.
    r^{2}-8r+25=0
    r=4+3i or r=4-3i
    y_{c}=c_{1} e^{4x}cos(3x)+c_{2}e^{4x}sin(3x)

    Now, to find the solution y=y_{c}+y_{p}, where y_{p} is the particular solution to the non-homogeneous equation, I need help. I guessed a few functions but none seemed to work out.

    Here is what Ive tried:
    Since 3e^{4x}cos(3x)=3e^{(4+3i)x}...well the real part I think
    and
    4+3i is a root of the homogenous equatoin, so I need multiply my guess for y_{p} by x

    From here I cannot figure out what function I need to use...
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  2. #2
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    Quote Originally Posted by snaes View Post
    I can change this equation, using Eulers method, and get the answers using sin and cos, but using complex numbers is apparently faster and we have to do it this way for my professor...so if you could help me figure this out I would appreciate it.

    y''-8y'+25y=3 e^{4x} cos(3x)

    HERE IS WHAT I DID:
    First, I'll solve the homogeneous case to find the complementary solution and the roots.
    r^{2}-8r+25=0
    r=4+3i or r=4-3i
    y_{c}=c_{1} e^{4x}cos(3x)+c_{2}e^{4x}sin(3x)

    Now, to find the solution y=y_{c}+y_{p}, where y_{p} is the particular solution to the non-homogeneous equation, I need help. I guessed a few functions but none seemed to work out.

    Here is what Ive tried:
    Since 3e^{4x}cos(3x)=3e^{(4+3i)x}...well the real part I think
    and
    4+3i is a root of the homogenous equatoin, so I need multiply my guess for y_{p} by x

    From here I cannot figure out what function I need to use...
    Since the complementary solution involves the RHS, you will need to guess a particular solution of the form

    y_p = A\color{red}x\color{black}e^{4x}\cos{3x} + B\color{red}x\color{black}e^{4x}\sin{3x}.
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  3. #3
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    Thanks, but is there any way to convert this into a "complex" valued expression. I need to use something like: a x e^{(4+3i)x}.

    [edit] for myhomework we are supposed to do each problem with real valued (sine and cosine) which i can do AND using some e^{a+bi} kind of form, which i cannot do. Any help would be great.
    Last edited by snaes; March 27th 2010 at 12:28 PM.
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