# Thread: Characteristics of quasilinear PDE

1. ## 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.

2. Originally Posted by Jose27
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.

3. Try the following combinations and see where that leads you

$\displaystyle x x_t + y y_t - u_t$

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