To solve the partial differential equation $x^2r - 2xs + t + q = 0$ using Monge's method, we need to analyze this PDE and solve it step by step. Monge’s method is typically applied to first-order PDEs, and here we have a second-order PDE. However, we can convert it into a system of first-order PDEs and then attempt to use Monge’s method.

Let’s break down the approach:

Step 1: Identify the PDE in the canonical form

We are given: $x^2 \frac{\partial^2 z}{\partial x^2} - 2x \frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} + q = 0$

Here, let's write this equation in terms of standard notation for Monge’s method: $x^2 r - 2x s + t + q = 0$ Where:

• $r = \frac{\partial^2 z}{\partial x^2}$,
• $s = \frac{\partial^2 z}{\partial x \partial y}$,
• $t = \frac{\partial^2 z}{\partial y^2}$,
• $q$ could be a function depending on $x, y, z, \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}$.

Step 2: System of first-order PDEs

We introduce new variables for the first-order derivatives:

• $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.

Then, the system of first-order PDEs becomes:

\frac{dz}{dx} &= p, \\ \frac{dz}{dy} &= q, \\ \frac{dp}{dx} &= r, \\ \frac{dp}{dy} &= s, \\ \frac{dq}{dx} &= s, \\ \frac{dq}{dy} &= t. \end{aligned}$$ The second-order PDE, $$x^2r - 2xs + t + q = 0$$, can now be interpreted as a constraint on these variables. ### Step 3: Monge's method Monge's method involves constructing a characteristic equation from the original PDE. For our equation: $$x^2r - 2xs + t + q = 0$$ We seek the characteristic curves by solving the system of differential equations: $$\frac{dx}{x^2} = \frac{dy}{-2x} = \frac{dz}{t + q}$$ This forms the set of characteristic equations we can now solve. ### Step 4: Solve the characteristic system 1. Solve $$\frac{dx}{x^2} = \frac{dy}{-2x}$$ to obtain the relationship between $$x$$ and $$y$$. 2. Solve for $$z$$ by integrating the corresponding equation $$\frac{dz}{t + q}$$. By solving these characteristic equations, you will obtain a general solution to the PDE. Would you like to proceed with solving the system further or clarify any steps? --- Here are 5 questions that could expand on this: 1. What are the characteristic equations in Monge’s method? 2. How can Monge’s method be adapted for non-homogeneous PDEs? 3. What are the applications of Monge’s method in modern physics or engineering? 4. How do boundary conditions influence the solution using Monge's method? 5. How does Monge’s method compare with other methods for solving PDEs, like the method of characteristics? **Tip:** Monge’s method is particularly useful for first-order PDEs, but it can be adapted to higher-order PDEs by reducing them into a system of first-order PDEs.