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

Using variation of parameters, find the general solutions of the differential equation

2. Relevant equations

y''' - 3''y + 3y' - y = e^{t}/ t

where e^{t}/t = g(t)

3. The attempt at a solution

I know how to solve these types of equations when its a second order, but I don't understand what to do for the particular solution since there are 3 solutions to the associated homogeneous equation, y1 = e^{t}, y2 = te^{t}, y3 = t^{2}e^{t}.

Usually I would just take the 2 solutions and compute the Wronskian, then use the formula where it's -y1*integral([y2*g(t)]/W)dt + y2*integral([y1*g(t)]/W)dt.

Since there are three solutions though, I don't understand how to solve it. My textbook uses a different method where they use something like v1'y1 + v2'y2 + v3'y3 = 0, v1'y1' + v2'y2' + v3'y3' = 0, and then the next equation is the same except the y's are the 2nd derivatives and it = g(t).

Then they solve for v1, v2 and v3, integrate, and plug them into the homogeneous equation to get the particular solution.

Sorry if this isn't clear!

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# Homework Help: Differential equations, variation of parameters

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