A characteristic equations is an equation of degree
For a differential equation
whose solutions
Derivation
Starting with a linear homogeneous differential equation with constant coefficients
Since
Formation of the General Solution
Solving the characteristic equation allow one to find the general solution of the differential equation. The solution depends on where the roots are at the complex frequency plane, and whether there are repeated roots.
Distinct Real Root
The superposition principle of a linear system says that if
Example: Solve
Repeated Real Root
If the characteristic equation has a root
Example
thus
Complex Root
What if a root
We can simplify it further with Euler’s formula:
Since
Note that both
Example:
Solve characteristic equation
Thus,
Applying initial values
, ,
Real Solutions
Sometimes we want to only find real solutions, in this case we can simplify the formula a bit further.
This yields
Relation with Eigenvectors and Eigenvalues
In linear algebra, we also have the characteristic equation in the form of
where
For a linear, homogeneous differential equation with constant coefficients:
The characteristic equation is
We can turn this nth order differential equation into a first-order system of differential equation by setting
This gives us:
For example. for the equation
The characteristic polynomial of this matrix is:
We get back the characteristic equation of the original differential equation (up to a multiplication of -1), demonstrating the direct link between the two concepts. 3
Footnotes
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- Linear Systems and Signals, 3rd Edition, 2.2
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What is the relation between the characteristic equation in linear algebra and the characteristic equation discussed in differential equations? - Quora ↩