When to ask the homogeneous question

[TG3]

When determining a particular solution to a differential equation, one of the necessary steps is to ask the "homogeneous question" aka Does any term in yp solve the homogeneous equation for this problem. When it does, I know that it is necessary to multiple by t.

My question is, do I multiply yp by T and then derive, or do I derive and then multiply by T?

For example, A cos(t) solves the homogeneous equation for a problem I'm working on. Do I derive it first:
y(p) = Acos(T)
Y'(p) = -Asin(T)
Y''(p) = -Acos(T)
And then multiply each of those by T, or do I multiply by T first and then derive? It makes a big difference!

 

[Mark44]

When determining a particular solution to a differential equation, one of the necessary steps is to ask the "homogeneous question" aka Does any term in yp solve the homogeneous equation for this problem. When it does, I know that it is necessary to multiple by t.

My question is, do I multiply yp by T and then derive, or do I derive and then multiply by T?

Try to keep your variables straight. t and T usually have different meanings.

For example, A cos(t) solves the homogeneous equation for a problem I'm working on. Do I derive it first:
y(p) = Acos(T)
Y'(p) = -Asin(T)
Y''(p) = -Acos(T)
And then multiply each of those by T, or do I multiply by T first and then derive? It makes a big difference!

The short answer is you multiply by t first.

If cos(t) is a solution to the homogeneous equation, then so is sin(t), so your particular solution can't be a linear combination of cos(t) and sin(t). In this case, your particular solution will include Atcos(t) + Btsin(t), unless of course, tcos(t) and tsin(t) happen to be solutions of the homogeneous equation. If that happens, then your particular solution will include t²cos(t) and t²sin(t).