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Use variation of parameters to find a general solution to the differential equation given that the functions y1 and y2 are linearly independ
Question
Use variation of parameters to find a general solution to the differential equation given that the functions y1 and y2 are linearly independent solutions to the corresponding homogeneous equation for t > 0.
ty” + (2t – 1)y’ – 2y = 7t2 e-2t y1 = 2t – 1, y2 = e-2t
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Mathematics
3 years
2021-07-29T01:44:48+00:00
2021-07-29T01:44:48+00:00 1 Answers
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Recall that variation of parameters is used to solve second-order ODEs of the form
y”(t) + p(t) y'(t) + q(t) y(t) = f(t)
so the first thing you need to do is divide both sides of your equation by t :
y” + (2t – 1)/t y’ – 2/t y = 7t
You’re looking for a solution of the form
where
and W denotes the Wronskian determinant.
Compute the Wronskian:
Then
The general solution to the ODE is
which simplifies somewhat to