Solving PDE for 2nd Order Conic Equation

[Bruno Tolentino]

I want to understand the solutions for the PDE of 2nd order $$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} f_x\\ f_y \end{bmatrix} +cf=0$$ But, this depends of the associated conic equation $$\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c=0$$ be a parbola, elipse or a hyperbola. One time that it already is known, I think that is necessary to simplify the equation eliminating terms through the change of coordinates.

Derivating the conic wrt x $$\frac{\partial }{\partial x}\left (\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c\right )= \frac{\partial }{\partial x}\left (0 \right )$$ and wrt y $$\frac{\partial }{\partial y}\left (\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix}:\begin{bmatrix} xx & xy \\ yx & yy \end{bmatrix} + \begin{bmatrix} b_1\\ b_2 \end{bmatrix}\cdot \begin{bmatrix} x\\ y \end{bmatrix} +c\right )= \frac{\partial }{\partial y}\left (0 \right )$$ obtains other two equations. These two news equations are straight and the intersection between they is the center of the conic of coordinates (β₁, β₂). So, the system (x, y) is relationed with the new system (x', y') by following vetorial equation $$\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix} \begin{bmatrix} x'\\ y' \end{bmatrix} + \begin{bmatrix} \beta_1\\ \beta_2 \end{bmatrix}$$

Now, is necessary know $$\begin{bmatrix} \alpha_{11} & \alpha_{12} \\ \alpha_{21} & \alpha_{22} \end{bmatrix}$$ How to determinate this matrix? Will be that the Mohr circle can help me?

 

[jvicens]

Thank you for your interest in understanding the solutions for the PDE of 2nd order. To answer your question, the matrix \begin{bmatrix}
\alpha_{11} & \alpha_{12} \\
\alpha_{21} & \alpha_{22}
\end{bmatrix} can be determined through a change of coordinates. This means that by substituting the variables x and y with new variables x' and y', we can simplify the equation and eliminate terms. This is known as a transformation of coordinates.

The Mohr circle is a graphical method used in mechanics and engineering to determine the state of stress at a point in a material. It is not directly related to the solutions of the PDE, but it may be helpful in understanding the behavior of the material under stress.

To determine the matrix \begin{bmatrix}
\alpha_{11} & \alpha_{12} \\
\alpha_{21} & \alpha_{22}
\end{bmatrix}, we can use the fact that the intersection of the two straight equations obtained by differentiating the conic equation will give us the center of the conic. This center can be expressed in terms of the new coordinates x' and y', and by equating it with the original coordinates x and y, we can find the values of the matrix elements.

I hope this helps in your understanding of the solutions for the PDE of 2nd order. If you have any further questions, please do not hesitate to ask.