A linear differential equation in two variables is an equation involving two dependent variables and their derivatives. It can generally be expressed in terms of $x$, $y$ (the dependent variables), and their partial derivatives.

The general form of a first-order linear partial differential equation (PDE) involving two variables, $u(x, y)$, is:

$A(x, y) \frac{\partial u}{\partial x} + B(x, y) \frac{\partial u}{\partial y} + C(x, y) u = D(x, y)$

where:

• $\frac{\partial u}{\partial x}$ is the partial derivative of $u$ with respect to $x$,
• $\frac{\partial u}{\partial y}$ is the partial derivative of $u$ with respect to $y$,
• $A(x, y), B(x, y), C(x, y), D(x, y)$ are given functions of $x$ and $y$,
• $u(x, y)$ is the unknown function to be solved.

This is a linear equation because the terms involving $u$ and its derivatives are of the first degree (i.e., no higher powers or products of the dependent variable and its derivatives).

Example:

Consider a simple linear PDE:

$\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$

This is a homogeneous linear differential equation in two variables.

Would you like more details on solving linear differential equations or a specific example?

Here are 5 related questions to explore this further:

1. What is the general solution to the equation $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0$?
2. How does the method of characteristics help solve linear PDEs?
3. How are boundary conditions used in solving partial differential equations?
4. What is the difference between homogeneous and non-homogeneous linear PDEs?
5. Can second-order linear PDEs be solved using the same techniques as first-order ones?

Tip: For solving PDEs, it's crucial to understand both the equation form and the boundary/initial conditions provided.