Diff EQ: Method of Undetermined Coefficients

[Saladsamurai]

Homework Statement

Solve using Method of Undetermined Coefficients:

y'' - 2y' + 3y = 15t - 8

The Attempt at a Solution

First I used the characteristic equation to solve for the homogeneous solution to get:

y_h = C₁e^-t + C₂e^3t

Great. Now I need to find the particular. I am suppose to assume that each particular solution takes the form of a linear combination of each forcing function in f(t) on the right hand side of the original DE: 15t - 8.

Now here is my question (it's kind of silly):

Do I use:

y_p1 = A*(15t - 8) +B*(d/dt)[15t - 8] and
y_p2 = 0

or

y_p1 = A*15t +B*(d/dt)[15t] and
y_p2 = C(8)

or does it matter? I feel like it should not matter. My book says that the right hand side (the forcing function) of a second order DE takes the form f(t) = f₁(t) + f₂(t)

When the terms are distinct as in f(t) = t³ + sin(t) it is easy to distinguish the f's --> f₁ = t³ and f₂ = sin(t).

But when it is polynomial: f(t) = at³ + bt² + ct +d
I feel like it should not matter how I break up f(t).

Any thoughts?

 

[fzero]

You're correct. You only need to test different types of particular solutions if the forcing function has different classes of functions.

 

[Saladsamurai]

You're correct. You only need to test different types of particular solutions if the forcing function has different classes of functions.

Ok great! Thank you I was trying to figure out if a solution I was getting was because of my assumed f's (e.g. the ones I made above) or because of an algebraic error. I am pretty sure I can chalk it up to the latter.

Thanks again.