Constant coefficient, Homogeneous equation, Homogeneous solution, Particular solution, Method of Undetermined Coefficients, Trial Functions Method, Quadratic polynomial, Exponential expression, Expression with sine or cosine, General solution, Resonance.
In cases where the particular solution includes exponential expressions, well, we could face some problems.
The next slideshow will discuss that.
If the particular solution includes an term, then we may face some problems when trying to solve it.
The first step is to solve this so called homogeneous equation:
Then we proceed to the particular solution.
We substitute this into the original equation:
Next, let's see the meaning of resonance.
This occurs when there is in the particular solution, and its exponent is exactly the same as the exponent of the homogeneous solution.
This time the exponents are not the same, so there is no resonance.
But now, there is.
Let’ see what happens now.
So, it is equal to the homogeneous solution.
This is what we call “resonance”.
And in this case an x comes here.
Let's see another one.
The homogeneous solution is the usual one:
The particular solution will include a linear expression,
and another where there is resonance.
First-order constant coefficient linear differential equations - The resonance
Calculus 2 episode