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Classify the equation $u_{xx} + 2u_{xy} = 0$, what are the characteristic curves and normal form?

Comparing the given pde with $A u_{xx} + Bu_{xy} +Cu_{yy} +H(u_x, u_y, x, y, u) =0 $ we get

$A = 1, \; B = 2, \; C=0$

So we have $B^2 - 4AC = 4 > 0$

So the given equation represents hyperbola. --- This is the classification $\ldots(1)$.

To find the characteristic curves, we need to solve

$A(\frac{dy}{dx})^2 -B(\frac{dy}{dx}) + C = 0 \rightarrow \frac{dy}{dx} = \frac{B - \sqrt{B^2 - 4AC}}{2A}, \frac{B + \sqrt{B^2 - 4AC}}{2A} = 0, 2 $

$$\rightarrow \frac{dy}{dx} = 0 \;\;\; \frac{dy}{dx} = 2\\ y = c_1 \;\;\; y = 2x + c_2 \\ y = c_1 \;\;\; y - 2x = c_2 \;\; \ldots \text{are the characteristic curves} \ldots (2)$$ Let $\xi = y \;\; \text{ and } \; \eta = y - 2x $

Then, $$u_x = u_\xi \xi_x + u_\eta \eta_x = -2u_\eta \rightarrow u_{xx} = -2(u_{\eta \xi} \xi_x + u_{\eta\eta}\xi_x) = 4u_{\eta\eta} $$

$$u_{xy} = -2(u_{\eta\xi}\xi_y + u_{\eta \eta}\eta_y) = -2(u_{\eta\xi} + u_{\eta \eta})$$

Hence, we get $$u_{xx} + 2u_{xy} = 0\\ 4u_{\eta\eta} -4 (u_{\eta\xi} + u_{\eta\eta}) = 0 \\ u_{\eta\xi } = 0 \;\;\ldots\text{is the normal form} \;\ldots(3)$$

Are my solutions in (1), (2), and (3) correct?

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What you did is correct but the final result is missing. $$u(x,y)=f(y)+g(y-2x)$$ with arbitrary functions $f$ and $g$.

Another method to find the above result :

$$u_{xx}+2u_{xy}=\frac{\partial}{\partial x}(u_x+2u_y)=0$$ Integrate : $$u_x+2u_y=\phi(y)\quad\text{any function}\quad \phi$$ First order linear PDE through method of characteristics : $$u(x,y)=\int \frac12 \phi(y)dy+g(y-2x)$$ Let $\frac12\int \phi(y)dy=f(y)$ $$u(x,y)=f(y)+g(y-2x)$$

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