Help need to solve Second order PDE

[Paras7]

Given the 2nd order linear PDE

\begin{align}
x^2u_{xx}-2xtu_{xt}+t^2u_{tt}+xu_x+tu_t &=0, & &x>0,\ t \in \mathbb{R} \tag 1
\end{align}

The principal part of the 2nd order equation is:
$$x^2u_{xx}-2xtu_{xt}+t^2u_{tt}$$
1.Where $$a=x^2,\ b=-xt,\ c=t^2$$So, the discriminant is
$$D=b^2-ac=(-xt)^2-x^2\cdot t^2=x^2t^2-x^2t^2=0$$
That means it has a parabolic form.

2. How does this equation look in canonical form?

We have to find the characteristics
$$\frac{dt}{dx}=\frac{b\pm \sqrt{D}}{a}\Leftrightarrow \frac{dt}{dx}=\frac{-xt}{x^2}=-\frac{t}{x}$$
and the solution is $$t=\frac{c}{x}\Leftrightarrow c=x \cdot t$$
We consider the transformation to be
\begin{align}
\xi&=xt\\
\eta&=x
\end{align}

So

$$J=\begin{vmatrix} \xi_x & \xi_t \\ \eta_x & \eta_t \end{vmatrix}=\begin{vmatrix} t & x \\ 1 & 0 \end{vmatrix}=x \neq 0$$
We know that $x>0$ by assumption.

Then the transformation is smooth because $x \neq 0$

We also know that $$u=u(\xi,\eta)=u(\xi(x,y),\eta(x,y))$$

Thus,

- $u_x=tu_{\xi}+u_{\eta}$
- $u_t=xu_{\xi}$
- $u_{x x}= t^2u_{\xi \xi}+t u_{\xi \eta}+tu_{\xi \eta}+u_{\eta \eta}$
- $u_{tt}=x^2u_{\xi \xi}$
- $u_{xt}=xtu_{\xi \xi}+xu_{\xi \eta}+u_{\xi}$

If we plug in all the above to the $(1)$ gives
$$x^2u_{\eta \eta}+u_{\eta}=0$$.

Is it possible something like that?


Could anyone verify the second-order partial derivative, please?