The given equation is a first-order partial differential equation:

$z = p^2 + q^2$ where $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$.

This is a Hamilton-Jacobi equation. Let's solve it using the method of characteristics:

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Step 1: Write the characteristic equations

The general form of a first-order PDE is: $H(x, y, z, p, q) = 0$ Here, $H(x, y, z, p, q) = z - p^2 - q^2$. The characteristic equations are:

\frac{dy}{dt} = \frac{\partial H}{\partial q}, \quad \frac{dz}{dt} = p \frac{\partial H}{\partial p} + q \frac{\partial H}{\partial q}.$$ For this PDE: $$dxTo solve the PDE $$z = p^2 + q^2$$, we use the **method of characteristics**. --- ### Step 1: Characteristic Equations The PDE can be expressed in terms of the **Hamiltonian**: \[ H(x, y, z, p, q) = z - p^2 - q^2.$$ The characteristic equations derived from $$H$$ are: $$\frac{dx}{dt} = \frac{\partial H}{\partial p}, \quad \frac{dy}{dt} = \frac{\partial H}{\partial q}, \quad \frac{dz}{dt} = p\frac{\partial H}{\partial p} + q\frac{\partial H}{\partial q}, \quad \frac{dp}{dt} = -\frac{\partial H}{\partial x}, \quad \frac{dq}{dt} = -\frac{\partial H}{\partial y}.$$ For $$H = z - p^2 - q^2$$: $$\frac{dx}{dt} = 2p, \quad \frac{dy}{dt} = 2q, \quad \frac{dz}{dt} = 2p^2 + 2q^2, \quad \frac{dp}{dt} = 0, \quad \frac{dq}{dt} = 0.$$ --- ### Step 2: Solve the Characteristic Equations 1. **$$dp/dt = 0$$ and $$dq/dt = 0$$:** - $$p$$ and $$q$$ are constants along the characteristic curves. Let $$p = p_0$$ and $$q = q_0$$. 2. **Solve for $$x$$ and $$y$$:** - From $$\frac{dx}{dt} = 2p$$, we get: $$x = x_0 + 2p_0t.$$ - From $$\frac{dy}{dt} = 2q$$, we get: $$y = y_0 + 2q_0t.$$ 3. **Solve for $$z$$:** - From $$\frac{dz}{dt} = 2p^2 + 2q^2$$, we get: $$z = z_0 + (2p_0^2 + 2q_0^2)t.$$ --- ### Step 3: Express the General Solution Using the initial conditions: $$x = x_0 + 2p_0t, \quad y = y_0 + 2q_0t, \quad z = z_0 + (2p_0^2 + 2q_0^2)t.$$ Let $$x_0$$ and $$y_0$$ be the parameters defining the family of characteristic curves. We can write the general solution as: $$z = \phi(x - 2p_0t, y - 2q_0t) + p_0^2 + q_0^2,$$ where $$\phi$$ is an arbitrary function determined by boundary/initial conditions. Let me know if you'd like further clarification or assistance with specific boundary conditions! --- ### Related Questions: 1. How is the method of characteristics derived for general PDEs? 2. What are Hamilton-Jacobi equations, and where are they applied? 3. How does the solution to this PDE change with specific boundary conditions? 4. Can the method of characteristics be used for second-order PDEs? 5. How do we visualize characteristic curves in this problem? --- ### Tip: Always check if a given PDE can be expressed in a Hamilton-Jacobi form, as it simplifies solving using well-established characteristic methods