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Thread: u_{xx}-u=0

  1. #1
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    u_{xx}-u=0

    Find the general solution to $\displaystyle u_{xx}-u=0$

    The solution is $\displaystyle \displaystyle u(x,y)=f(y)e^x+g(y)e^{-x}$

    How is this obtained?

    Thanks.

    I solved the first part:

    $\displaystyle \displaystyle P(x)=-1\Rightarrow e^{\int P(x)dx}=e^{-\int dx}=e^{-x}$

    $\displaystyle \displaystyle \int\frac{\partial}{\partial x}\left[e^{-x}u_x\right]dx=\int 0dx\Rightarrow e^{-x}u_x(x,y)=f(y)$

    Should that be in the integral the partial derivative or just derivative?

    $\displaystyle \displaystyle\Rightarrow u_x(x,y)=e^xf(y)$
    Last edited by dwsmith; Dec 23rd 2010 at 03:45 PM.
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  2. #2
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    I think of this problem as find the solution $\displaystyle u(x,y)$ such that $\displaystyle u(x,y) = u_{xx}$

    As you only have a partial with respect to the same variable then the solution is similar to that of a 2nd order ODE. I.e $\displaystyle y'' = y$ has the solution $\displaystyle y=e^x$ and $\displaystyle y= e^{-x}$

    The difference in your problem is that $\displaystyle f(y)$ and $\displaystyle g(y)$ need to be introduced but are constant with respect to $\displaystyle x$
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  3. #3
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    Quote Originally Posted by dwsmith View Post
    $\displaystyle \displaystyle P(x)=-1\Rightarrow e^{\int P(x)dx}=e^{-\int dx}=e^{-x}$
    Why not try separation of variables?
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  4. #4
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    Quote Originally Posted by pickslides View Post
    I think of this problem as find the solution $\displaystyle u(x,y)$ such that $\displaystyle u(x,y) = u_{xx}$

    As you only have a partial with respect to the same variable then the solution is similar to that of a 2nd order ODE. I.e $\displaystyle y'' = y$ has the solution $\displaystyle y=e^x$ and $\displaystyle y= e^{-x}$

    The difference in your problem is that $\displaystyle f(y)$ and $\displaystyle g(y)$ need to be introduced but are constant with respect to $\displaystyle x$
    I just updated my post as you replied.

    My problem is should it be partial derivatives or written like an ODE with just the derivative?

    Should I use the same method for the integration again?
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  5. #5
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    Quote Originally Posted by pickslides View Post
    Why not try separation of variables?
    That was what I thought of when the light bulb turned on.
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    I understand your method which is simple but I would like to finish the problem how I started it. If you wouldn't mind, how should I finish it off?

    Thanks.

    If I just integrate, I wouldn't obtain the e^{-x} so I am not sure how to proceed.
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  7. #7
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    Quote Originally Posted by dwsmith View Post
    I understand your method which is simple but I would like to finish the problem how I started it. If you wouldn't mind, how should I finish it off?

    Thanks.

    If I just integrate, I wouldn't obtain the e^{-x} so I am not sure how to proceed.
    Well your attempt suggests finding an integrating factor in a 2nd order DE, will that work? Should you not be finding a characteristic equation?

    ;D
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  8. #8
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    I just looked it up. Integrating factors aren't restricted to first order linear equations.
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    I would say $\displaystyle \displaystyle u_{xx}-u(x,y)=0$

    Gives characteristic equation in the form $\displaystyle \displaystyle r^2-1= (r-1)(r+1)=0 \implies r_{1,2}=\pm 1 $ and $\displaystyle \displaystyle y=c_1e^{r_1x}+c_2e^{r_2x}$ therefore seek a solution in the form of $\displaystyle \displaystyle y=c_1e^x+c_2e^{-x}$

    As we have a PDE $\displaystyle \displaystyle c_1$ and $\displaystyle \displaystyle c_2$ can also be separate functions of $\displaystyle \displaystyle y$.
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  10. #10
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    Now, I am looking to find the sol. of $\displaystyle u_{xx}-u=0$ which satisfies the auxiliary conditions

    $\displaystyle u(0,y)=\varphi (y) \ , \ u_x(0,y)=\psi (y)$

    $\displaystyle u(0,y)=f(y)+g(y)=\varphi (y)$

    $\displaystyle u_x(0,y)=f(y)-g(y)=\psi (y)$

    How do I incorporate this into the solution?

    $\displaystyle u(x,y)=e^xf(y)+e^{-x}g(y)$
    Last edited by dwsmith; Dec 23rd 2010 at 05:04 PM.
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  11. #11
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    Is this the correct procedure?

    $\displaystyle u(0,y)=f(y)+g(y)=\varphi (y)$

    $\displaystyle u_x(0,y)=f(y)-g(y)=\psi (y)$

    By adding the two equations and simplifying, we obtain $\displaystyle \displaystyle f(y)=\frac{\varphi (y)+\psi (y)}{2}$

    Now, solving for $\displaystyle g(y)$ I get $\displaystyle \displaystyle g(y)=\frac{\varphi (y)-\psi (y)}{2}$

    $\displaystyle \displaystyle u(x,y)=e^x\left[\frac{\varphi (y)+\psi (y)}{2}\right]+e^{-x}\left[\frac{\varphi (y)-\psi (y)}{2}\right]$
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