Question

Reduce into canonical form
Uxx+Uxy+Uyy+Ux=0

Answer 1

$\Delta = a_{12}^2 - {a_{11}}{a_{22}} = {(\frac{1}{2})^2} - 1 \cdot 1 = - \frac{3}{4} < 0$ - we have an equation of elliptic type.

Characteristic equation:

$\frac{{dy}}{{dx}} = \frac{{{a_{12}} + \sqrt \Delta }}{{{a_{11}}}} = \frac{{\frac{1}{2} + \frac{{\sqrt 3 }}{2}i}}{1} \Rightarrow y = \frac{x}{2} + \frac{{\sqrt 3 }}{2}ix + {C_1}$

or

$\frac{{dy}}{{dx}} = \frac{{{a_{12}} - \sqrt \Delta }}{{{a_{11}}}} = \frac{{\frac{1}{2} - \frac{{\sqrt 3 }}{2}i}}{1} \Rightarrow y = \frac{x}{2} - \frac{{\sqrt 3 }}{2}ix + {C_2}$

Let

$\xi (x,y) = \frac{{{C_1} + {C_2}}}{2} = \frac{1}{2}\left( {y - \frac{x}{2} - \frac{{\sqrt 3 }}{2}ix + y - \frac{x}{2} + \frac{{\sqrt 3 }}{2}ix} \right) = y - \frac{x}{2}$

$\eta \left( {x,y} \right) = \frac{{{C_2} - {C_1}}}{{2i}} = \frac{1}{{2i}}\left( {y - \frac{x}{2} + \frac{{\sqrt 3 }}{2}ix - y + \frac{x}{2} + \frac{{\sqrt 3 }}{2}ix} \right) = \frac{{\sqrt 3 }}{2}x$

$u(x,y) = v\left( {\xi ;\,\eta } \right)$

Then

${u_x} = {v_\xi }{\xi _x} + {v_\eta }{\eta _x} = - \frac{1}{2}{v_\xi } + \frac{{\sqrt 3 }}{2}{v_\eta }$

${u_y} = {v_\xi }{\xi _y} + {v_\eta }{\eta _y} = {v_\xi }$

${u_{xx}} = - \frac{1}{2}\left( {{v_{\xi \xi }}{\xi _x} + {v_{\xi \eta }}{\eta _x}} \right) + \frac{{\sqrt 3 }}{2}\left( {{v_{\eta \xi }}{\xi _x} + {v_{\eta \eta }}{\eta _x}} \right) = - \frac{1}{2}\left( { - \frac{1}{2}{v_{\xi \xi }} + \frac{{\sqrt 3 }}{2}{v_{\xi \eta }}} \right) + \frac{{\sqrt 3 }}{2}\left( { - \frac{1}{2}{v_{\eta \xi }} + \frac{{\sqrt 3 }}{2}{v_{\eta \eta }}} \right) = \frac{1}{4}{v_{\xi \xi }} - \frac{{\sqrt 3 }}{2}{v_{\xi \eta }} + \frac{3}{4}{v_{\eta \eta }}$

${u_{yy}} = {v_{\xi \xi }}{\xi _y} + {v_{\xi \eta }}{\eta _y} = {v_{\xi \xi }}$

${u_{yx}} = {v_{\xi \xi }}{\xi _x} + {v_{\xi \eta }}{\eta _x} = - \frac{1}{2}{v_{\xi \xi }} + \frac{{\sqrt 3 }}{2}{v_{\xi \eta }}$

Substitute the found values ​​into the original equation:

$\frac{1}{4}{v_{\xi \xi }} - \frac{{\sqrt 3 }}{2}{v_{\xi \eta }} + \frac{3}{4}{v_{\eta \eta }} - \frac{1}{2}{v_{\xi \xi }} + \frac{{\sqrt 3 }}{2}{v_{\xi \eta }} + {v_{\xi \xi }} - \frac{1}{2}{v_\xi } + \frac{{\sqrt 3 }}{2}{v_\eta } = 0$

${v_{\xi \xi }}\left( {\frac{1}{4} - \frac{1}{2} + 1} \right) + \frac{3}{4}{v_{\eta \eta }} + {v_{\xi \eta }}\left( { - \frac{{\sqrt 3 }}{2} + \frac{{\sqrt 3 }}{2}} \right) - \frac{1}{2}{v_\xi } + \frac{{\sqrt 3 }}{2}{v_\eta } = 0$

$\frac{3}{4}{v_{\xi \xi }} + \frac{3}{4}{v_{\eta \eta }} = \frac{1}{2}{v_\xi } - \frac{{\sqrt 3 }}{2}{v_\eta }$

${v_{\xi \xi }} + {v_{\eta \eta }} = \frac{2}{3}{v_\xi } - \frac{{2\sqrt 3 }}{3}{v_\eta }$

Answer: ${v_{\xi \xi }} + {v_{\eta \eta }} = \frac{2}{3}{v_\xi } - \frac{{2\sqrt 3 }}{3}{v_\eta }$ , where $\xi (x,y) = \ y - \frac{x}{2}$, $\eta \left( {x,y} \right) = \frac{{\sqrt 3 }}{2}x$ , $u(x,y) = v\left( {\xi ;\,\eta } \right)$