Homogenous Equations with Constants
Coefficients
In a homogenous equation, coefficient are constant
Reduction of Order
Second Order Homogeneous equations can be formatted into several First Order Linear equations using reduction of order
Example
Consider the equation
Exponential Solutions
Assume the solutions are exponentials
Given that the Wronskian
The solutions to the equation can be found by finding the solutions to
Example
Consider the same equation
Example
Consider the equation and initial condition
Complex Roots
If the detriment
To solve complex roots the initial set up remains the same
If
Apply Euler's formula
However these solutions are still complex, to find real solutions use the Superposition Principle
Example
Repeated Roots
If two solutions are equivalent
So therefore
Apply Variation of Parameters
Since
Example
Variation of Parameters
For a non characteristic equation, assume that the solution is
Therefore
Example
Consider the equation
The equation can not be solved via Reduction of Order
| Reduction of Order | Variation of Parameters | |
|---|---|---|
| Method | Using several lower order equations to solve a high level equation |
Requires one or more homogenous solutions Substitute coefficients to cancel out the zeroth order term |
| Example | Ex1 3.1 | HW1 Last Problem Ex4 3.4 |