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Math Help - Some more Differential Equations

  1. #1
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    Some more Differential Equations

    The question states to find the solution of the following -

     \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 10y = 20 - e^2x

    I found the root, it was 3+- i

    Im not too sure what to do after this, any help would be appreciated.
    Thanks in advance
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  2. #2
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    Hello, brd_7!

    Find the solution: .  \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 10y \;= \;20 - e^{2x}

    I found the root, it was 3 \pm i

    I'm not too sure what to do after this.

    Those roots give us the homogeneous solution: . y \;=\;e^{3x}\left(C_1\cos x + C_2\sin x\right)


    Now we must find the particular solution . . .

    I conjectured that the solution is of the form: . y_p \:=\:A + Be^{2x}

    . . and found it to be: . y_p \;=\;2 - \frac{1}{2}e^{2x}


    The solution is: . y \;=\;e^{3x}\left(C_1\cos x + C_2\sin x\right) + 2 - \frac{1}{2}e^{2x}


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  3. #3
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    Quote Originally Posted by brd_7 View Post
    [snip]
     \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 10y = 20 - e^2x

    I found the root, it was 3+- i

    Im not too sure what to do after this, any help would be appreciated.
    [snip]
    Then you need to read this.
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  4. #4
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    Ok, i managed to prove it all myself.. I just didnt know what to do because of the extra '20' term on the RHS.. now i know that you just put an A to represent the constant makes it a whole lot easier. Thank you!
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