Definitions of Differential Equations
Setting the stage: What is a DE, what is an "Order", and what makes a DE "Linear"?
Introduction
A Differential Equation (DE) is an equation containing one or more derivatives of an unknown function. Instead of finding a number (like ), our goal is to find a function that satisfies the relationship.
They are the language of utility in physics, engineering, and biology, describing how things change over time.
Key Terms
Order
The Order of a DE is the order of the highest derivative in the equation.
Example: is 2nd Order because of .
Linearity
A DE is Linear if the unknown function and its derivatives appear to the first power only and are not multiplied together or inside other functions (like sin(y)).
Form: .
Visual: Family of Solutions
Interactive: General Solution
The General Solution includes an arbitrary constant C. An Initial Value (like y(0)=1) picks exactly one curve.
Worked Examples
Example 1: Identify Order and Linearity
Classify the following differential equations:
A)
2nd Order, Linear. The coefficients () depend only on t. y and y'' are linear.
B)
1st Order, Non-Linear. The term makes it non-linear.
C)
3rd Order, Linear.
Example 2: Verifying a Solution
Verify that is a solution to .
1. Find derivatives:
2. Plug into LHS:
3. Conclusion:
LHS = RHS. It is a solution.
Example 3: Initial Value Problem (IVP)
Solve with .
1. Integrate to find General Solution:
2. Apply Initial Condition:
Set :
3. Particular Solution:
Practice Quiz
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