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# Find the type, transform to normal form, and solve. Show your work in detail.$u_{xx}-6u_{xy}+9u_{yy}=0$

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The coefficients of the equation

$u_{xx} -6y_{xy}+ 9u_{yy} =0$

are

$A=1, \, B=-3, \, C=9 .$

Since

$AC - B^2 = 9-9=0$

we see that this is $\textbf{parabolic}$ type equation. The charateristic equation we solve is

$(y^\prime)^2 +6y^\prime + 9 =0 .$

which can be rewritten as

$(y^\prime +3)^2 =0 .$

It is clear that its solutions are

$y^\prime =-3$

which leads to the differential equation

$dy =-3 dx.$

Integrating we get

$y=-3x.$

So the $\textbf{characteristic(s)}$ is (are)

$\boxed{ \Phi (x,y) = \Psi (x,y) = y+3x}$

So we introduce new variables $v= v$ and $w= y+3x$. We have that

\begin{aligned}u_x & = u_v \cdot \frac{\partial v}{\partial x} + u_w \cdot \frac{\partial w}{\partial x} = u_v +3u_w, \\ u_{xx}& =\frac{\partial}{\partial x} u_v +3 \frac{\partial }{\partial x} u_w =\\ & = \left( u_{vv} \frac{\partial v}{\partial x} +u_{vw} \frac{\partial w}{\partial x}\right) +3 \left( u_{wv} \frac{\partial v}{\partial x} +u_{ww} \frac{\partial w}{\partial x}\right) = \\ & = ( u_{vv} +3 u_{vw}) +3 ( u_{vw} +3 u_{ww} ) = \\ & = u_{vv} +6 u_{vw} +9 u_{ww} \\ u_{y} & = u_v \cdot \frac{\partial v}{\partial y} + u_w \cdot \frac{\partial w}{\partial y} =u_w, \\ u_{yy} &= \frac{\partial }{\partial y} u_w = u_{ww}, \\ u_{xy} & = \frac{\partial }{\partial y} u_x = \frac{\partial }{\partial y} (u_v +3 u_w) = u_{vw} +3 u_{ww} \end{aligned}

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