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Math Help - Differential equation

  1. #1
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    Differential equation

    Hello everyone! (Merry Christmas and happy holidays)

    I got to solve

    -y'' + y = 0

    I'm so lost.
    I tried cos(x) and sin(x)

    then i tried y = sin(-x) => y' = -cos(-x) => y'' = -sin(-x)

    =>  -y'' + y = -(-sin(-x))+ sin(-x) = sin(-x)+sin(-x) = -2 sin(x) \not= 0

    The solution is not that obvious, I guess.

    Does anyone knows the solution?

    Kind regards,
    Rapha
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  2. #2
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    The DE can be rearranged to the form

     \frac{d^2y}{dx^2}=y

    what is a function that when differentiated twice gives the function itself

    Spoiler:

    y=e^x

    you can also solve this by separating variables twice
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  3. #3
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    Quote Originally Posted by Rapha View Post
    Hello everyone! (Merry Christmas and happy holidays)

    I got to solve

    -y'' + y = 0

    I'm so lost.
    I tried cos(x) and sin(x)

    then i tried y = sin(-x) => y' = -cos(-x) => y'' = -sin(-x)

    =>  -y'' + y = -(-sin(-x))+ sin(-x) = sin(-x)+sin(-x) = -2 sin(x) \not= 0

    The solution is not that obvious, I guess.

    Does anyone knows the solution?

    Kind regards,
    Rapha
    The DE has the general form a \frac{d^2 y}{dx^2} + b \frac{dy}{dx} + c y = 0.

    The standard technique is to try a solution of the form y = A e^{\lambda x}.

    Read these:

    PlanetMath: second order linear differential equation with constant coefficients

    Homogeneous 2nd-order differential equations

    to get started.
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  4. #4
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    Hello, Rapha!

    This is one of the most basic forms . . .


    Solve: . -y'' + y \:=\:0
    We have: . y'' - y \:=\:0


    Let: y \:=\:e^{mx}\quad\Rightarrow\quad y' \:=\:me^{mx}\quad\Rightarrow\quad y'' \:=\:m^2e^{mx}

    . . Substitute: . m^2e^{mx} - e^{mx} \:=\:0

    . . Divide by e^{mx}\!:\;\;m^2 - 1 \:=\:0 \quad\Rightarrow\quad m \:=\:\pm1

    . . Hence, the solutions are: . y \:=\:e^x,\;\;y \:=\:e^{-x}


    Form a linear combination of the solutions: . y \;=\;C_1e^x + C_2e^{-x}

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  5. #5
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    y''-y=0

    The characteristic equation is: y^2-1=0

    y_1=-1
    y_2=1

    The solution is y=C1*e^{x*y_1}+C2*e^{x*y_2}

    y=C1*e^{x}+C2*e^{-x}
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