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Math Help - another differential equation

  1. #1
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    another differential equation

    i need help with this one
    solve
     y'' + 9y = 0; \quad y = 3, \; y'=3 \;\textnormal{at}\; x = \pi/3

     y(x) = ?
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  2. #2
    Senior Member tukeywilliams's Avatar
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    Write down the characteristic equation. Then see what form the solution has. So  r^2 + 9r = 0 or  r(r+9) = 0 , which means  r = 0, - 9 which are real solutions. Then the general solution has the form  y = c_{1}e^{r_{1}t} + c_{2}e^{r_{2}t} . So  y' = c_{1}r_{1}e^{r_{1}t} + c_{2}r_{2}e^{r_{2}t} . You are given the initial conditions.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by tukeywilliams View Post
    Write down the characteristic equation. Then see what form the solution has. So  r^2 + 9r = 0 or  r(r+9) = 0 , which means  r = 0, - 9 which are real solutions. Then the general solution has the form  y = c_{1}e^{r_{1}t} + c_{2}e^{r_{2}t} . So  y' = c_{1}r_{1}e^{r_{1}t} + c_{2}r_{2}e^{r_{2}t} . You are given the initial conditions.
    The characteristic equation here is r^2+9=0, so r=\pm ~ 3 \bold{i}

    RonL
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  4. #4
    Senior Member tukeywilliams's Avatar
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    Whoops. Sorry about that. In that case the general solution would be  y(t) = c_{1}e^{(a+\bold{i}b)t} + c_{2}e^{(a - \bold{i}b)t} .
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  5. #5
    MHF Contributor red_dog's Avatar
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    When r=a\pm bi, the general solution is
    y=e^{ax}(C_1\cos bx+C_2\sin bx)
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by tukeywilliams View Post
    Whoops. Sorry about that. In that case the general solution would be  y(t) = c_{1}e^{(a+\bold{i}b)t} + c_{2}e^{(a - \bold{i}b)t} .
    Quote Originally Posted by red_dog View Post
    When r=a\pm bi, the general solution is
    y=e^{ax}(C_1\cos bx+C_2\sin bx)
    These are the same thing in different notation.

    RonL
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  7. #7
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    im a little confused on applying the conditions
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  8. #8
    MHF Contributor red_dog's Avatar
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    y(x)=C_1\cos 3x+C_2\sin 3x
    y'(x)=-3C_1\sin 3x+3C_2\cos 3x
    Now, put x=\frac{\pi}{3} and solve the system
    \left\{\begin{array}{ll}y\left(\frac{\pi}{3}\right  )=3\\y \ '\left(\frac{\pi}{3}\right)=3\end{array}\right.
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