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Math Help - Find the general solution of the given differential equation

  1. #1
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    Find the general solution of the given differential equation

    y''+y=7sin(2t)+5tcos(2t)

    Since g(t) is a sum I'll have 2 equations.

    y''+y=7sin(2t)

    y''+y=5tcos(2t)

    I'm kind of confused on which forms of Y to use.

    Would Y_1 (t)=Asint(2t) and Y_2 (t)=(At+B)sin(2t)+(At+B)cos(2t) ?
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  2. #2
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    Re: Find the general solution of the given differential equation

    No, you don't need to do two equations.

    Have you evaluated the homogeneous equation's solution yet?
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  3. #3
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    Re: Find the general solution of the given differential equation

    Do you mean y(t)=y_c (t)+ Y(t) ?

    If so, y(t) = c_1 cos(t)+c_2 sin(t) + Y(t)

    I can't find Y(t) though.
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  4. #4
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    Re: Find the general solution of the given differential equation

    For "linear differential equations with constant coefficients" if the "right hand side" functions are those that we expect as solutions to a homogeneous equation, e^{ax}, sin(x), cos(x), powers of t, and products of those, then you can expect such functions as solutions of the equation.

    Here, the right side is 7sin(2t)+ 5t cos(2t). So try a solution of the form y= Asin(2t)+ Bcos(2t)+ Ctsin(2t)+ Dtcos(2t).
    Then y'= 2A cos(2t)- 2B sin(2t)+ Ct cos(2t)+ C sin(2t)- Dt sin(2t)+ D cos(2t)= (Ct+ 2A+ D)cos(2t)+ (C-2B- Dt) sin(2t) and
    y''= C cos(2t)- 2(Ct+ 2A+ D)sin(2t)- D sin(2t)+ 2(C- 2B- Dt) cos(2t).

    Put those into the differential equations to get equations to solve for A, B, C, and D.
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  5. #5
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    Re: Find the general solution of the given differential equation

    I was able to get that A=11/9, B=0, C=0, and D=-5/3.

    But based off of my answer choices A should equal either 1/9 or -7/9.
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  6. #6
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    Re: Find the general solution of the given differential equation

    HallsofIvy made a calculation error or two.

    \begin{align*}y(t) & = A\sin(2t) + B\cos(2t) + Ct\sin(2t) + Dt\cos(2t) \\ y'(t) & = 2A\cos(2t) - 2B\sin(2t) + 2Ct\cos(2t) +C\sin(2t) -2Dt\sin(2t) + D\cos(2t) \\ & = (2A + D)\cos(2t) + (C-2B)\sin(2t) + 2Ct\cos(2t) - 2Dt\sin(2t) \\ y''(t) & = -2(2A+D)\sin(2t) + 2(C-2B)\cos(2t) -4Ct\sin(2t)+2C\cos(2t) -4Dt\cos(2t) -2D\sin(2t) \\ & = (4C-4B-4Dt)\cos(2t)-(4A+4D+4Ct)\sin(2t) \\ y(t) + y''(t) & = (4C-3B-3Dt)\cos(2t) - (3A + 4D + 3Ct)\sin(2t) \end{align*}

    So 5t = 4C-3B-3Dt and 7 = -(3A+4D+3Ct). Hence B = C = 0, D = -\dfrac{5}{3}, and A = -\dfrac{1}{9}.

    That's closer to what you were expecting...
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  7. #7
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    Re: Find the general solution of the given differential equation

    Thanks to you all.

    So do I ever need to have 2 or more individual equations when g(t) is a sum of equations to find the particular solution? I have a picture to show what I mean.
    Attached Thumbnails Attached Thumbnails Find the general solution of the given differential equation-screen-shot-2013-10-17-2.33.20-pm.png  
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  8. #8
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    Re: Find the general solution of the given differential equation

    One of these days I really have to learn arithmetic!
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  9. #9
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    Re: Find the general solution of the given differential equation

    You can solve them all at once or one at a time, however you prefer.
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  10. #10
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    Re: Find the general solution of the given differential equation

    Ok, thanks again to you all
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