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Differential Equations/3rd order linear ODE

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Question
Hi,
I'm having much trouble solving this 3rd order linear ODE.  It is

y''' - 8y = 0

I get the auxiliary/characteristic equation of

r^3 - 8 = 0

From that I get

r = 2

Which leads me to the solution of

y = c1*e^2x + c2*x*e^2x + c3*x^2*e^2x

I double checked my work and found that this is not the solution.  So, I used a program called Scientific Workplace to solve it.  It gave me

y = c1*e^2x + c2*e^-x*cos(sqrt(3)x) + c3*e^-x*sin(sqrt(3)x)

I have no idea of where the root of -1 +/- i*sqrt(3) comes from.  I fail to see where I'm going wrong here.  I've used this method to solve other 3rd order linear ODEs, but can't here.  Can you tell me where I'm going wrong or if I can't use this method for this ODE?  Thanks!

Answer
I kept thinking the answer of mine needed fixing, and finally saw how to really solve the ODE.
I get something similar to what they get.
Try it this way and see what you get.

If we let the solution be f(x) = c(e^-x)(sin(ax) + cos(ax)), we get
f'(x) = c(e^-x)[(-1-a)sin(ax) + (-1+a)cos(ax)].  This leads to
f"(x) = c(e^-x)[(1+2a-a²)sin(ax) + (1-2a-a²)cos(ax)].  Differentiating again gives
f"'(x) = c(e^-x)[(-1+3a+3a²-a)sin(ax) + (-1+3a+3a²-a)cos(ax)].
That leads to f"'(x) = c(e^-x)(-1+3a+3a²-a³)[sin(ax)+cos(ax)].

To solve this, so -1+3a+3a²-a³ = 0.  As can be seen, -1 is a root, so we can factor out (1+a).
Factoring gives us (1+a)(-1+4a-a²).  That is the same as -(1+a)(1-4a+a²).
The roots of the quadratic are [4ħsqrt(16-4)]/2 = [4ħsqrt(12)]/2 [4ħ2*sqrt(3)]/2 = 2ħsqrt(3).
That reduces to 2ħsqrt(3), which is 1ħsqrt(3).

Thus, we have the roots a = -1, 2+sqrt(3), and 2-sqrt(3).
Putting these in and using the equations sin(a+b) = sin(a)cos(b) + cos(a)sin(b),
sin(a-b) = sin(a)cos(b) - sin(b)cos(a), cos(a+b) = cos(a)cos(b) - sin(a)sin(b), and
cos(a-b) = cos(a)cos(b) + sin(a)sin(b) may give the something similar to the answer suggested.

After doing all this, I just realized I forgot that there was a 2 in the exponential.
Try modifying what I gave with that in mind and right back again with further questions ...
if necessary.

Differential Equations

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Scott A Wilson

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I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

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I have taken Differential Equations at OSU. Occasionally I have assisted people with questions on the subject.

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I wrote a thesis on the study of shock waves and rarefaction fans. That occurs in nature when mathematics fails to apply. See, mathematics says that when shock waves appear, there should be two solutions whereas nature knows there is only one way to be in that area. When rarefaction fans occur, the mathematics says there should be no solution, but despite this, nature still moves ahead with its own plan.

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I received an MS degree at OSU in Mathemematics and a BS degree at OSU in Mathematical Science.

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I graduated from OSU with a Bachelor of Science degree in mathematics with honors. Two years later, I graduated from OSU with a Master of Science degree in mathematics with honors. Between these two degrees, I have more hours in graduate courses that required to get them.

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I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR, as Safeway in Washington, and many other areas. I have answered over 8,500 right here, but only a little over 190 of them have been in differential equations.

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