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Math Help - find the general solution ODE pls help

  1. #1
    tak
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    find the general solution ODE pls help

    d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x

    please check if my answer are correct

    y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by tak View Post
    d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x

    please check if my answer are correct

    y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x
    Hello tak,
    You can substitute y back in the differential equation and see if it holds.This way you can be more independent(you can even do this in exams )

    So try it
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  3. #3
    tak
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    does it mean i have to difficentiate again

    Does it mean I have to differentiate again?
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  4. #4
    Behold, the power of SARDINES!
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    Quote Originally Posted by tak View Post
    d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x

    please check if my answer are correct

    y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x
    I think we may have a problem.

    \frac{d^2y}{dx}-\frac{dy}{dx}-y=3e^{-x}+10\sin(x)-4x

    solving the homogenious equation for the particular solution we get..

    \frac{d^2y}{dx}-\frac{dy}{dx}-y=0

    m^2-m-1=0 \iff m^2-m+\frac{1}{4}=1+\frac{1}{4} \iff (x-\frac{1}{2})^2=\frac{5}{4} \iff x=\frac{1 \pm \sqrt{5}}{2}

    so

    y_c=c_1e^{\frac{1+\sqrt{5}}{2}x}+c_2e^{\frac{1-\sqrt{5}}{2}x}

    See what you can do from here.
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  5. #5
    tak
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    sorry it should be 2y

    \frac{d^2y}{dx}-\frac{dy}{dx}-2y=3e^{-x}+10\sin(x)-4x<br />
    very sorry it should be 2y
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  6. #6
    Behold, the power of SARDINES!
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    Quote Originally Posted by tak View Post
    \frac{d^2y}{dx}-\frac{dy}{dx}-2y=3e^{-x}+10\sin(x)-4x<br />
    very sorry it should be 2y
    then the equation is

    m^2-m-2=0 \iff (m-2)(m+1)

    so

    y_c=c_1e^{2x}+c_2e^{-x}

    Now we need to find the particular solution

    since e^{-x} in the complimentry solution
    the particular solution will be of the form

    y_p=\underbrace{Ax^2+Bx+C}_{forThe-4x}+\underbrace{E\cos(x)+F\sin(x)}_{forThe10\sin(x  )}+\underbrace{G(e^{-x})+H(xe^{-x})}_{forTheRepeted 3 e^{-x}}
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  7. #7
    tak
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    multiplixity one

    Hi

    I Know that i need to multiply by x but do I need to keep the existing term for the last part.
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  8. #8
    tak
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    last part

    how do you know that you need to keep the G(e^-x) term. I saw some website they don't add in and some does. Can I know th rule
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