3.1. Linear and Non-linear Partial Differential Equations

Try these first

1. 1.

   Define what a Partial Differential Equation (PDE) is.

   Hint

   Think about how it differs from ordinary differential equations.

2. 2.

   What is the general form of a linear PDE?

   Hint

   Remember that the dependent variable and its derivatives should be in first power only.

3. 3.

   What is a partial differential equation (PDE)?

   Hint

   Recall the definition we discussed.

4. 4.

   Which of the following is a characteristic of a linear PDE?

   • A) It involves higher powers of variables.
   • B) Variables can be multiplied together.
   • C) Dependent variables appear to the first power.

   Hint

   Think about the structure of linear equations.

5. 5.

   Prove that the equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ is a linear PDE, and provide a physical interpretation of its meaning.

   Hint

   Check the way terms are structured; focus on their power.

6. 6.

   Create a real-life scenario where a non-linear PDE would be used, and explain the significance of being non-linear in that context.

   Hint

   Consider the nature of fluid motion and challenges in modeling it accurately.