(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I seem to not fully understand how to find particular solutions. I'm having a hard time guessing what the solutions maybe. I'll explain as I write out the problem.

[tex]y''' - 2y'' + y' = te^t + t^2 + 4[/tex], find the general solution.

2. Relevant equations

[tex]Z(r) = r^3 - 2r^2 + r[/tex]

3. The attempt at a solution

Found the homogenous solution:

[tex]r^3 - 2r^2 + r = 0[/tex]

[tex]r (r - 1)^2 = 0[/tex]

[tex]r_1 = 0, \ r_2 = 1, \ r_3 = 1[/tex]

[tex]y = C_1 + C_2e^t + C_3te^t[/tex]

Now I'm trying to find the particular solutions.

For [tex]te^t[/tex], I assume:

[tex]y_1 = At^2e^t[/tex]

[tex]y_1' = Ae^t (t^2 + 2t)[/tex]

[tex]y_1'' = Ae^t (t^2 + 4t + 2)[/tex]

[tex]y_1''' = Ae^t (t^2 + 6t + 6)[/tex]

Now I plug back in:

[tex]Ae^t (t^2 + 6t + 6) -2Ae^t (t^2 + 4t + 2) + Ae^t (t^2 + 2t) = te^t[/tex]

[tex]2A = t[/tex]

[tex]A = \frac{t}{2}[/tex]

[tex]y_1 = \frac{1}{2}t^3e^t[/tex]

Now, for [tex]t^2[/tex], I assumed [tex]y_2 = Bt^3[/tex], but in an example in the book that had [tex]t^2[/tex], it said that [tex]y_2 = t^2 (A_1t^2 + A_2t + A_3)[/tex]. I don't understand how they got that.

Also, for the constant 4, I assumed [tex]y_3 = Ct[/tex]. Is this correct?

I don't fully understand how I am supposed to go about guessing the solutions. There must be some guidelines, but we weren't taught what they are. If someone could help me with this, it would be greatly appreciated.

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# Finding a particular solution (Diff. Eq.)

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