What is a Differential Equation?
An equation containing the derivatives of one or more unknown functions (or dependent variables), with respect to one more independent variables, is a differential equation (DE).
Why are Differential Equations important?
They often represent physical quantities and their rate of change, giving us a relationship to predict physical events.
What terminology is used?
Differential Equations: Equations involving derivatives that describe how one quantity changes relative to another.
Order: The highest derivative present; e.g., a second derivative indicates a second-order equation.
Types:
- Ordinary Differential Equations (ODEs): Involve functions of a single variable and their derivatives.
- Partial Differential Equations (PDEs): Involve functions of multiple variables and their partial derivatives.
Forms:
- Normal Form: A standard, simplified expression of a differential equation.
- Differential Form: Representation in terms of derivatives.
Linearity:
- Linear Differential Equations: The dependent variable and its derivatives appear to the first power and are not multiplied together.
- Non-Linear Differential Equations: The dependent variable or its derivatives appear in powers greater than one or are multiplied together.
How do we use this terminology to classify Differential Equations?
- Type: is it an ordinary or partial differential equation?
- Order: what is the highest order derivative?
- Linearity: is it linear or non-linear?
How do we verify solutions to Differential Equations?
Substitute the solution into the differential equation, differentiating it where required.
EXAMPLE
Verify whether is a solution to the differential equation .
The solution to this example is like so:
The differential equation in question is an ordinary first order linear differential equation.