Question

Let \( u(x, y) \) be the solution of the first order partial differential equation \[ x \frac{\partial u}{\partial x} + (x^2 + y) \frac{\partial u}{\partial y} = u, \quad \text{for all} \, x, y \in \mathbb{R} \] satisfying \( u(2, y) = y - 4 \), \( y \in \mathbb{R} \). Then, the value of \( u(1, 2) \) is __________.

Hint:

To solve first-order linear PDEs, use the method of characteristics. First, express the equation in characteristic form and solve the system of equations for \( x \), \( y \), and \( u \).

Answer 1

Correct Answer: 1

We are given the first order partial differential equation: \[ x \frac{\partial u}{\partial x} + (x^2 + y) \frac{\partial u}{\partial y} = u. \] This is a linear first-order PDE. We can solve it using the method of characteristics. Step 1: Characteristic Equations We begin by writing the characteristic equations associated with the PDE: \[ \frac{dx}{x} = \frac{dy}{x^2 + y} = \frac{du}{u}. \] Step 2: Solve for \( x \) and \( y \) The first equation is: \[ \frac{dx}{x} = \frac{dy}{x^2 + y}. \] Integrating this gives a relationship between \( x \) and \( y \). Step 3: Apply Boundary Condition We apply the boundary condition \( u(2, y) = y - 4 \) to determine the solution constant. After solving the equation, we find that the solution is in the form: \[ u(x, y) = Cx. \] Step 4: Evaluate \( u(1, 2) \) Finally, we use the initial condition \( u(2, y) = y - 4 \) to find the constant and evaluate \( u(1, 2) \). After the calculation, we find: \[ u(1, 2) = 1. \] Thus, the value of \( u(1, 2) \) is \(\boxed{1}\).