Linear Differential Equations
A systematic method for solving equations of the form using a magical multiplier.
Introduction
A First Order Linear DE can always be written in the standard form:
The trick to solving these is to multiply the entire equation by a special function called an Integrating Factor, which collapses the left side into a single derivative using the Product Rule.
The Method
Recipe
- Standardize: Divide by coefficient of to get .
- Calculate Factor: . (Ignore here).
- Multiply: Multiply standard form by .
The LHS becomes . - Integrate: .
- Solve: Divide by to isolate .
Visual: Transient Terms
Interactive: Converging Solutions
For , the integrating factor is . Solutions are . Notice how the initial condition eventually doesn't matter.
Worked Examples
Example 1: Basic Application
Solve .
1. Integrating Factor:
. So .
2. Multiply:
Multiply original DE by :
.
LHS collapses: .
3. Integrate:
.
4. Solve for y:
Multiply by :
.
Example 2: Variable Coefficients
Solve , for .
1. Standard Form (Important!):
Divide by t: .
2. Integrating Factor:
.
3. Multiply:
Multiply by standard form:
.
4. Integrate:
.
5. Final Solution:
.
Example 3: Initial Value Problem
Solve , .
1. Factor:
.
2. Integrate Product:
.
Use integration by parts (twice) for RHS: .
.
.
3. Initial Condition:
.
.
Practice Quiz
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