Mathematics: Find the general solution of y + 6y + y − 34y = 0 if it is known that y1 = e−4x cos(x) is one solution.

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Find the general solution of y + 6y + y − 34y =

alyssamonae

02.01.2022

Step-by-step explanation:

Given is a differential equation of III order,

y''' + 6y'' + y' - 34y = 0

The characteristic equation would be cubic as

m^3+6m^2+m-34=0

By trial and error, we find that

$f(2) = 2^3+6(2^2)+2-34 =0\\$

Thus m=2 is one solution

Since given that $e^{-4x} cos x$ is one solution we get

m = -4+i and hence other root is conjugate m=-4-i

Hence general solution would be

$y=Ae^{2x} +e^{-4x} (Bcosx +C sinx)$

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