Suppose that a fourth order differential equation has a solution y=8e^(2x)xsinx.
(a) Find such a differential equation, assuming it is homogeneous and has constant coefficients.
(b) Find the general solution to this differential equation. In your answer, use A,B,C and D to denote arbitrary constants and x the independent variable.
Find a differential equation?
2015-02-23 6:09 pm
回答 (1)
2015-02-23 6:39 pm
the 8 is going to be one of our ABCD constants...
y = A e^2x x sin x + B e^2x sin x + C e^2x x cos x + D e^2x cos x
what would be the characteristic equation to give you something like this?
you have roots (r - 2 + i)(r - 2 - i) = 0... this will give you
y = B e^2x sin x + D e^2x cos x
but in order to get the e^2x x sin x you have double roots....
[(r - 2 + i)(r - 2 - i)]^2 = 0
(r^2 - 4r + 5)^2
r^4 - 8 r^3 + 26r^2 - 40 r + 1 = 0
y'''' - 8 y''' + 26 y'' - 40 y' + y = 0
y = A e^2x x sin x + B e^2x sin x + C e^2x x cos x + D e^2x cos x
what would be the characteristic equation to give you something like this?
you have roots (r - 2 + i)(r - 2 - i) = 0... this will give you
y = B e^2x sin x + D e^2x cos x
but in order to get the e^2x x sin x you have double roots....
[(r - 2 + i)(r - 2 - i)]^2 = 0
(r^2 - 4r + 5)^2
r^4 - 8 r^3 + 26r^2 - 40 r + 1 = 0
y'''' - 8 y''' + 26 y'' - 40 y' + y = 0
收錄日期: 2021-04-15 18:24:52
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