📚 Topic Summary
Partial Differential Equations (PDEs) involve functions of several variables and their partial derivatives. Understanding the order, linearity, and homogeneity of a PDE is crucial for selecting appropriate solution methods. The order is the highest order derivative in the equation. Linearity means the equation is a linear combination of the function and its derivatives. Homogeneity refers to whether the equation equals zero when the function and its derivatives are all zero.
📝 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Order |
A. A PDE where the dependent variable and its derivatives appear linearly. |
| 2. Linear PDE |
B. A linear PDE where, if $u$ is a solution, so is $cu$ for any constant $c$. |
| 3. Homogeneous PDE |
C. The highest order derivative present in the PDE. |
| 4. Partial Derivative |
D. A derivative of a function of several variables with respect to one of those variables. |
| 5. PDE |
E. An equation involving an unknown function of several variables and its partial derivatives. |
(Match the numbers 1-5 with letters A-E)
✍️ Part B: Fill in the Blanks
Complete the following sentences using the words provided: linear, order, partial, homogeneous, derivatives.
A Partial Differential Equation (PDE) involves _______________ _______________ of an unknown function. The _______________ of a PDE is determined by the highest _______________ present in the equation. A PDE is _______________ if the unknown function and its _______________ appear linearly, and it is _______________ if setting all functions to zero results in zero.
🤔 Part C: Critical Thinking
Consider the following PDE: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x,y)$. Explain how the nature of $f(x,y)$ affects whether this PDE is homogeneous or non-homogeneous. Give examples.