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# Variation of Parameters Question (simple)?

I need help with this Variation of Parameters question.

The question is y"+2y'+y=e^-t . I know the solution to the homogeneous portion, but when I try to solve for the particular solutions, I don't know what step to take. For variation of parameters I need to find two functions that fit the equation y"+2y'+y=0 and so far i only have e^-t as a function. What is the first step to solve for the non-homogeneous portion of the problem. Thanks!

### 1 Answer

- kbLv 78 years agoFavorite Answer
Homogeneous solution:

y'' + 2y' + y = 0 has characteristic equation r^2 + 2r + 1 = (r + 1)^2 = 0.

==> r = -1, -1.

So, y_h = Ae^(-x) + Bxe^(-x).

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Now, we can use variation of parameters.

Assume that y = e^(-x) u + (xe^(-x)) v for some functions u(x) and v(x).

See if you can take it from here...