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Thread: Characteristics of quasilinear PDE

  1. #1
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    Characteristics of quasilinear PDE

    Ok, so I've been trying to solve this for a while now and I don't seem to get anywhere. Hope someone can point in the right direction.

    We have the PDE

    $\displaystyle x(y^2+u)u_x-y(x^2+u)u_y=(x^2-y^2)u$

    where $\displaystyle x,y$ are independent variables and $\displaystyle u=u(x,y)$.

    So, I know that the equations for the characteristics in this case are

    $\displaystyle x_t=x(y^2+u)$

    $\displaystyle y_t=-y(x^2+u)$

    $\displaystyle u_t=(x^2-y^2)u$

    but, I don't see how I can obtain $\displaystyle t,s$ from here, I've tried playing with the coefficients in the original, isolating $\displaystyle u$ or $\displaystyle u_x$ but that gets me nowhere. I'm thinking of proposing the curve $\displaystyle x^2-y^2=c$ as a characteristic, but I haven't checked if it is. So, if someone could give any hints as to how to solve this it would be appreciated.
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  2. #2
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by Jose27 View Post
    Ok, so I've been trying to solve this for a while now and I don't seem to get anywhere. Hope someone can point in the right direction.

    We have the PDE

    $\displaystyle x(y^2+u)u_x-y(x^2+u)u_y=(x^2-y^2)u$

    where $\displaystyle x,y$ are independent variables and $\displaystyle u=u(x,y)$.

    So, I know that the equations for the characteristics in this case are

    $\displaystyle x_t=x(y^2+u)$

    $\displaystyle y_t=-y(x^2+u)$

    $\displaystyle u_t=(x^2-y^2)u$

    but, I don't see how I can obtain $\displaystyle t,s$ from here, I've tried playing with the coefficients in the original, isolating $\displaystyle u$ or $\displaystyle u_x$ but that gets me nowhere. I'm thinking of proposing the curve $\displaystyle x^2-y^2=c$ as a characteristic, but I haven't checked if it is. So, if someone could give any hints as to how to solve this it would be appreciated.
    Have you considered the proportion ratios

    $\displaystyle \frac{dx}{x(y^2+u)}-\frac{dy}{y(x^2+u)}=\frac{du}{u(x^2-y^2)}(=dt)$

    Where you try to eliminate an independent variable.
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  3. #3
    MHF Contributor
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    Try the following combinations and see where that leads you

    $\displaystyle x x_t + y y_t - u_t$

    $\displaystyle u\left(y x_t + x y_t\right) + x y u_t$.
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